Nicky Maynard reduced the arrears with a cracking volley while Bristol City were, for long periods, the better side.
But Hart, who became Rangers' ninth manager in little over two years when he replaced Jim Magilton last week, now has a draw and a win to show from his first two games in charge.
The hosts were slow starters as they struggled to get to grips with a new diamond midfield formation and were camped in their own half early on.
Hungarian star Akos Buzsaky, involved for the first time since his dressing-room 'difference of opinion' accounted for Magilton, was a bystander as City dominated.
Gavin Williams forced a save from home keeper Radek Cerny and Danny Haynes had a shot just deflected wide.
But, totally against the run of play, Rangers took the lead just after the half-hour when Fitz Hall crossed towards Simpson.
The on-loan Arsenal forward's initial touch was poor but the ball bounced kindly off Williams back into his path to poke beyond Dean Gerken for his ninth goal of the season.
City boss Gary Johnson had considered not travelling to the game due to a chest infection.
And he must have been regretting his decision not stay in his sick bed when, five minutes before the break, Rangers went two ahead.
Leigertwood collected Buzsaky's flick, cut inside on his left foot and curled an unstoppable effort from 20 yards low inside Gerken's right-hand post.
Simpson had a chance to put the hosts out of sight after the interval when Wayne Routledge sent him through but the young striker curled his shot high and wide.
And Johnson's mood improved in the 58th minute when Maynard conjured up his 11th goal of the season in superb style.
The striker juggled Williams' pass twice in turning and shaking off two Rangers defenders before his third touch, a stunning 20-yard volley, flew past Cerny and into the top corner.
Johnson and City were twice screaming for penalties in the last 10 minutes when Maynard tumbled under last-ditch challenges from Hall and Peter Ramage but referee David Phillips waved away the appeals.
Hart saw fit to sacrifice Simpson and Buzsaky for defenders Gary Borrowdale and Matt Connolly to hold off City's late flurry - prompting derision from a number of hard-to-please Rangers fans.
And the home boss breathed a sigh of relief when City sub David Clarkson hit the side netting with the visitors' last chance.