Sheridan only took over as manager at Rodney Parade on Friday after Terry Butcher's departure the previous day.
He saw his side fall behind to Exeter when Joel Grant took Jordan Moore-Taylor's cross and cut inside to score.
But Newport fought back as John-Lewis pounced after Troy Brown blocked Alex Rodman's initial effort.
Newport County manager John Sheridan told BBC Wales Sport: "We played very well, I couldn't have asked any more from the players.
"The endeavour and some of the football we played at times was really good.
"It's always difficult when you're bottom of the league, you're always looking up at everyone and it was a difficult game against Exeter, I've always liked the way they play.
"But the players really responded to the new manager coming in and they gave me a performance I was really pleased with."

Shenzhen prosecutors, late on Thursday, said a dispatcher and supervisor of the landfill, the chief and deputy manager of a company in charge of it, and seven other people were arrested.
They said they have been charged with negligently causing a serious accident.
The landslide happened when a hill of construction waste collapsed, leaving 74 people either missing or dead.
The formal arrests come three days after the police took what Chinese media called "coercive measures" against 12 people. It is thought the 11  arrested today were among those.
Officials have described the 20 December disaster as man-made, raising the possibility of severe punishments for those they consider responsible.
One official, whose job involved regulating construction sites, apparently jumped to his death a week after the landslide. It is not clear if he was being investigated for his role in it.
Officials have appealed for police and the public to help them find other suspects apparently on the run.

The House of Lords heard the government had not yet taken a decision as to whether an inquiry will be held.
Launching one now could prejudice the on-going investigations into the Hillsborough disaster, it heard.
About 10,000 strikers and 5,000 police officers clashed at the coking plant near Rotherham in June 1984.
More than 120 officers and pickets were injured and 93 people arrested.
On Tuesday, the Independent Police Complaints Commission (IPCC) ruled a full report into allegations of police misconduct will not be made public.
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Lord Richard Keen, Lords spokesperson for the Home Office, said a barrister had been commissioned to go through 10,000 police documents "in the context of the investigation at Orgreave".
In response to a question from Lord Richard Balfe, he said: "The IPCC has told the Home Office officials that if it announces any action to set up an inquiry or other investigation relating to Orgreave it would have an impact on the Hillsborough investigation.
"For that reason, the decision will only be taken after that part has been concluded."
He added work was still on-going to "assess whether material related to the policing of Orgreave is relevant to the Hillsborough criminal investigations".
Barbara Jackson, from the Orgreave Truth and Justice Campaign, said: "We respect all that the Hillsborough campaigners have had to go through, but we would like our issue dealt with as quickly as possible, as soon as their verdict is in the public domain."
New boss John Sheridan saw Newport claim a hard-fought draw in his first game in charge thanks to a Lenell John-Lewis equaliser against Exeter.
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Eleven people in China have been arrested for their alleged role in the Shenzhen landslide disaster.
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Any inquiry into police actions during the Battle of Orgreave would not take place until Hillsborough investigations conclude, the Home Office said.