Roberto Martinez was unmoved. Chomping on a piece of gum, arms folded tightly, Everton’s manager looked like a man who was feeling the tension of the day. The scene that was unfolding around him, however, was completely at odds with his demeanour. In one corner of Goodison Park, Ross Barkley was being mobbed by a group of ecstatic team-mates, while the stands were abuzz with exultant supporters. Barkley had just gracefully and gleefully ensured Everton had equalled their biggest win of a bewildering Barclays Premier League campaign, a 3-0 slaying of a wretched Newcastle United team who finished with 10 men, but, for a second, Martinez appeared reluctant to embrace the moment. James McCarthy (right) opened the the scoring for Everton on Sunday at Goodison Park and was congratulated by Aaron Lennon . Romelu Lukaku, having set up McCarthy for the opener, netted Everton's second from the penalty spot after Lennon was brought down . Ross Barkley celebrates after giving gloss to the victory with a stoppage time strike to make it 3-0 to the home side at Goodison Park . Everton (4-3-3): Howard 7; Coleman 7, Jagielka 7.5, Alcaraz 6.5, Baines 7; McCarthy 8 (Besic 86), Gibson 8, Osman 8.5; Lennon 7.5 (Atsu 83mins), Lukaku 8 (Barkley 74, 7), Kone 7. Unused subs: Garbutt, Naismith, Robles, Stones. Booked: Lennon. Manager: Roberto Martinez 8. Newcastle United (4-2-3-1): Krul 4; Janmaat 5, Coloccini 3, Williamson 5,  Taylor 5; Colback 5 Gouffran 4; Obertan 4 (Perez 46, 5), Sissoko 5; Ameobi 4 (Cabella 58, 5); Riviere 5 (Gutierrez 62, 6). Unused subs: Anita, Armstrong, Elliot, Satka. Booked: Taylor, Sissoko. Sent off: Coloccini. Manager: John Carver 6. Referee: Martin Atkinson 7. Man of the match: Leon Osman. Tim Krul was at fault for the opener, click here for more from our brilliant Match Zone. Soon, though, the magnitude of the result dawned and Martinez was smiling. This was a big victory. The spectre of a relegation battle has been tormenting Everton during what has been a wretched 2015 but here was a scoreline and a performance to ease tensions. James McCarthy set the ball rolling, the excellent Romelu Lukaku secured the win from the penalty spot and Barkley provided the gloss when waltzing on to Christian Atsu’s pass in injury time. True, Newcastle’s resistance was meek but here was a significant step in the right direction. ‘We performed with real desire,’ said Martinez. ‘I thought we were very much in control, and most important of all we played well and wanted to win. In the past we have had a fearful approach in not wanting to lose but we showed what we can do. We dealt with the extra pressure.’ Big efforts on a Thursday night have tended to zap Everton on a Sunday and a return of eight points from a possible 24 immediately after Europa League games has shown what kind of impact progress in that competition has had domestically. Significantly, Martinez was able to shuffle his pack and inject some freshness with five changes to the team that had beaten Dynamo Kiev; it worked perfectly. The return of Darron Gibson, Leon Osman and Aaron Lennon, especially, made Everton a much more vibrant proposition. Sammy Ameobi breaks away from McCarthy down the wing with the two sides goalless at Goodison Park during Sunday's first half . Ameobi fights back gallantly as the winger attempted to stop Everton's on-loan winger Lennon in his tracks . McCarthy fires in his shot from the edge of the area as Fabricio Coloccini chases, but the effort found the net and eased Everton nerves . Finding form in the Barclays Premier League has been difficult for the Toffees who showed their relief in the goal celebrations . Roberto Martinez and John Carver share a laugh on the touchline as their two teams do battle on Merseyside . Jonas Gutierrez, who was again a Newcastle substitute, goes in for a challenge on McCarthy in midfield as Newcastle chase the game . A side with more class than Newcastle might have made things more problematic for Everton and it would have been interesting to see what had happened if either Gabriel Obertan or Mike Williamson had turned in the respective chances that came their way in the second minute. Yet that was as good as it got from Newcastle. Acting manager John Carver argued that his side finished strongly but the reality was they were comfortably beaten and had they returned to Tyneside having lost by six, there could not have been any argument. As soon as McCarthy opened the scoring, the outcome was never in doubt, the Republic of Ireland wrong-footing Tim Krul from 20 yards after Lukaku had laid Phil Jagielka’s long-ball into his path. Krul, really, should have done better but Everton were not complaining. It was, incredibly, McCarthy’s first shot on target in the Premier League for 10 months but it could not have come at a better time, as the relief around Goodison was almost tangible. This stadium can be unforgiving when things are going badly but here they were ready to enjoy themselves. Being unable to get a second goal before the interval caused a few murmurs but, with Newcastle’s play becoming more abject by the minute, there was never any sense Everton would get pegged back and they ended the contest in the 56th minute. Referee Martin Atkinson had no hesitation pointing to the spot after Yoan Gouffran got into a tangle and upended Lennon and Lukaku did the rest, rolling his kick into the Gwladys Street, just as he had done against Kiev. The Belgian had taken 20 penalties in training on Wednesday and scored 17. Newcastle’s task was made impossible when Coloccini was dismissed for clattering into Lennon – Carver said he would appeal if he felt Newcastle had a case but when he studies the video, he will realise an appeal would be futile – and that prompted the visiting fans to again barrack Mike Ashley. It was another one of those days for Newcastle but Carver, to his credit, would not throw in the towel. Lukaku steps up to slot home his second penalty of the week after also netting in the Europa League on Thursday night at Goodison Park . Tim Krul dived the wrong way and Lukaku simply had to find the net with his spot-kick, which he duly did to double the lead . The big Belgian forward celebrates after effectively sealing Everton's first win at Goodison Park of the calendar year on Sunday afternoon . Coloccini lunges in on Lennon within three minutes of Everton's second goal and is shown the red card by referee Martin Atkinson . Mike Williamson slides in to deny Arouna Kone as Everton surged forward in search of a third goal against their 10-man opponents . Barkley rounds Krul during stoppage time to add Everton's third after an impressive performance from the substitutes' bench . ‘There was a lot of disappointment after the game, usually, we leave them for a couple of days, but I asked the guys if they had anything to say, and two or three did,’ said Carver. ‘It was sensible, I won't name names, but they had something to say. There weren't tea cups thrown, they were sensible, and examined it well. I can't be down. I have never thrown the towel anywhere, and I won't do it here. I want them to be positive.’ On this evidence, that will prove easier said than done. The England youngster angles in his shot as Newcastle's defenders and the rest of Goodison Park can only watch on as it happens . Barkley celebrates his strike with the Merseyside crowd who were delighted to see their side win at home for the first time this year . The Newcastle players applaud the travelling fans who have hung around after the final whistle despite their heavy defeat at Goodison .
Everton were just three points above the relegation zone at the start of play after Burnley's win on Saturday . James McCarthy wrong-footed Tim Krul with his low shot from just outside the area to give Everton 20th-minute lead . Romelu Lukaku doubled the lead with a penalty on 56 minutes after Aaron Lennon was fouled by Yoan Gouffran . Fabricio Coloccini was red carded for lunge at Lennon in the 59th minute as Newcastle fell apart . Arouna Kone rounded Newcastle keeper Krul in the closing stages but could not find the net from a tight angle . Ross Barkley capped the win with his second goal of the season during stoppage time with a cool finish .