The Sheffield side had been kept in the game by on-loan keeper Paul Gerrard before taking the lead with a dubious penalty decision in the 42nd minute.
Cameroon international Patrick Suffo, discarded by United boss Neal Warnock two seasons ago, produced a spectacular equalizer in the 67th minute before Lester stepped off the bench to snatch a United win.
Coventry soaked up early United pressure and caught them on the break in the 26th minute. Gary McAllister fired in a low strike from 15 yards that was snatched by the diving Gerrard.
Gerrard, signed on loan from Everton after Paddy Kenny's injury at Palace in midweek, foiled the Midlands side again three minutes later.
Midfielder Michael Doyle burst through minutes later as Gerrard pulled off another stunning stop before United took the lead following a dubious decision by a referee David Crick.
Peter Ndlovu burst into the box surrounded by three Coventry players before he tumbled over without any contact being made. Yazid Mansouri and Calum Davenport were booked before Michael Brown sunk his spot-kick into the left-hand corner.
City's assistant manager Erik Black was sent to the stand as he continued the argument.
Suffo struck the equaliser with a fericious left-foot shot, the ball crashing against the underside of the bar as it zipped past Gerrard, before Lester nodded the winner in the 87th minute running on to a Phil Jagielka cross to power a header in at the far post.