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."
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.