Liverpool boss Brendan Rodgers vowed not to give up on Champions League qualification after a convincing 2-0 win over Newcastle. The Reds moved within four points of the Barclays Premier League top four after goals from Raheem Sterling and Joe Allen saw off the struggling Magpies at Anfield. With fading champions Manchester City now looking vulnerable in fourth, Liverpool's hopes - which took a hefty blow in back-to-back losses to Manchester United and Arsenal - could have been reignited. Raheem Sterling wheels away in celebration after scoring Liverpool's opener . Rodgers said: 'The players and myself are very motivated to do the best we can. We have shown we can put results together consistently. We have important players coming back which is good news. 'But we are looking for other teams to slip up and we have to continue winning. 'There are teams above us but, as we have seen last year, there can be changes and points dropped and I think there will be changes.' Sterling curls the ball in to the far corner of Tim Krul's net to give Liverpool the lead . Liverpool, inspired by Philippe Coutinho, started superbly with Sterling scoring after nine minutes and several other chances being created. Newcastle rallied before the break and had a strong penalty appeal turned down but Allen made the game safe in the second half. Rodgers said: 'It was a good performance. I thought that our attacking play was very bright, we looked exciting going forward, we worked hard in the midfield. Sterling's shot nestles in to the back of the net as Krul picks himself up from Anfield turf . 'At times our possession could have been better but we had a real attacking intent and created chances and could have scored more goals.' The defeat was Newcastle's fifth in succession and left them with work still to do to ensure they do not get pulled into the relegation battle. Their misery was compounded when Moussa Sissoko was sent off for what was deemed a second yellow card offence, when he caught Lucas Leiva on the ankle. Manager John Carver admitted Sissoko could easily have been shown a straight red, however. Allen secured three points for Liverpool that narrowed the gap on Manchester City . He said: 'The sending off, it could have been a straight red, I will be honest. I have said that to Moussa, I am quite straight and I won't defend anyone if it is indefensible.' Asked if Sissoko could be fined, Carver said: 'I am sure there will be some kind of disciplinary procedure. I said before the game we needed to keep 11 players.' While Carver did not complain about that decision, he was angry that Ayoze Perez did not earn a penalty after apparently being felled by Dejan Lovren. He said: 'The game hinged on a huge decision. I don't want to be seen to be a moaner but it is fact we don't get big decisions for whatever reasons, it could be me. I have spoken to the referee, but it doesn't help me.' Lovren clears the ball with his head as Lucas and Perez also scrap for the ball . Carver did see reason for some optimism, feeling that his side's performance was much improved from their derby loss to Sunderland last time out. He said: 'I thought the first nine minutes we were very tentative, nervous after our performance against Sunderland. It was a bad goal to concede. 'After that we grew into the game. It is easy to crumble at Anfield but we didn't. One or two guys rolled their sleeves up. 'I think we got a response. We didn't get a result but I saw enough. 'There are lot of positives to take from the game. A lot of people expected us to get a hiding and I think we held our own for the majority of the game.'
Liverpool beat Newcastle 2-0 in the Premier League on Monday night . Raheem Sterling and Joe Allen saw off the Magpies at Anfield . Liverpool are four points behind fourth-placed Manchester City . READ: Jordan Henderson hopes Liverpool can turn up heat on Man City . READ: Rodgers will talk to Raheem Sterling about his behaviour .