But Richard Cresswell levelled for the Blades and Loftus Road's latest new boss was denied a winning start when Ben Watson headed against the crossbar in injury time.
Hart's appointment, following Jim Magilton's controversial exit this week, is the ninth managerial change in just over two years since owner Flavio Briatore arrived at Loftus Road.
Of the last three permanent bosses, Paulo Sousa lasted the longest with 26 matches at the helm, Magilton managed 23 and Iain Dowie just 15.
Little wonder, then, that Hart has only been given a contract until the end of the season.
United, by contrast, have had three managers in 10 years and, having resisted the temptation to sack Kevin Blackwell during a recent poor run, have now gone seven games unbeaten.
Hart received a lukewarm reception from a crowd clearly tiring of welcoming incoming managers, but the proceedings livened up considerably with Leigertwood's early strike.
The midfielder, who has played under all six permanent bosses and three caretakers since arriving from the Blades in 2007, lashed Watson's half-cleared corner through a crowd of players and, via a hefty deflection, past Mark Bunn.
The lead lasted just six minutes, though, after Watson's short backpass let in Toni Kallio who squared the ball for Cresswell to bundle over the line.
Both teams had chances thereafter but Bunn and fellow goalkeeper Radek Cerny barely had a save to make.
Bunn did halt Routledge in his tracks and Adel Taarabt hit the side netting for Rangers while United defender Matt Kilgallon headed over.
United should have gone in ahead in first-half stoppage time but Jamie Ward volleyed wide from eight yards out.
In the second half Routledge put a volley into the side netting, before Jay Simpson claimed a penalty, to no avail, after tripping over Chris Morgan's leg.
At the other end Quinn shot wide, James Harper blazed over and Cresswell's header fell the wrong side of the post.
And Hart missed out on a victorious start to his stint in west London when Watson headed against the bar at the death.