Hart sacrificed attacking duo Simpson and Akos Buzsaky for defenders Gary Borrowdale and Matt Connolly late on, to jeers from some hard-to-please Rangers fans.
But after registering his first victory in his second game in charge, he said: "We hadn't won in six games and lost the lead in our last two, and there was an edginess in our game against a very good Bristol City side.
"We are fighting for any sort of rhythm but each player showed the right attitude to get a result. This club needed a win today."
Hart is Rangers' ninth manager in little under two years as the club's mega-rich owners look for a quick route to promotion.
"I've only been here two weeks, I don't know people, I don't know the supporters," he added.
"I try not to let that sort of thing influence what I do.
"With the things that have happened at this club over the past few years, I think it is understandable that expectations are raised.
"When you've been in football a long time, then managing people's expectations seems to be as important as managing the team these days."
Rangers struggled to get to grips with Hart's new midfield diamond in the opening half an hour, but they went ahead totally against the run of play.
Fitz Hall's cross rebounded off Gavin Williams into the path of on-loan Arsenal forward Simpson, who turned the ball past Dean Gerken.
Five minutes before the break 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.
After the break Maynard conjured up his 11th goal of the season in superb style, juggling Williams' pass twice in turning and shaking off two Rangers defenders before his third touch, a stunning 20-yard volley, flew into the top corner.
City could have had two penalties when Maynard tumbled under last-ditch challenges from Hall and Peter Ramage.
But boss Gary Johnson refused to blame the decisions from referee David Phillips for his side's defeat.
"My only gripe is that the referee looked at his linesman with a question mark on his face, and the linesman looked back with the same question mark and so they just played on," he said. "And that happened twice. Neither made that decision.
"I could say we were robbed, but we have to do more as a team to compete in the top half of this division."