Substitute Hitchcock tapped into an empty net after Charlie Austin's effort had been parried by Ipswich goalkeeper Scott Loach.
Harry Redknapp's side had already hit the bar twice, first through Clint Hill and then Austin, before the 20-year-old son of QPR goalkeeping coach Kevin Hitchcock struck.
However, Rangers will be concerned about Junior Hoilett - a reported transfer target for Stoke - after he limped off with what looked like a hamstring injury seven minutes into the second half.
But the result maintains their encouraging start to life in the Sky Bet Championship following last season's relegation from the top flight.
They enjoyed a decent spell of pressure midway through the first half, during which Shaun Wright-Phillips' deflected shot went narrowly over and Nedum Onuoha missed the target with a header.
Austin, making his full home debut for the R's, was twice denied a goal in the space of a couple of minutes.
Hoilett found space on the left and picked out the former Burnley forward, whose first-time effort was blocked by Town defender Tommy Smith.
Then, Austin's shot from Danny Simpson's right-wing cross was well saved by Loach.
But Ipswich, who had won on four of their previous seven visits to Shepherd's Bush, had the best opportunity to go ahead before the break.
Jay Tabb sent in a cross in from the left and, after centre-back Richard Dunne challenged David McGoldrick, the loose ball fell to Daryl Murphy, whose weak attempt from eight yards out was tipped away by keeper Robert Green.
The loss of Hoilett was a blow for Rangers but they continued to press and almost went ahead shortly after the hour mark.
Joey Barton's free-kick from the right was headed against the frame of the goal by Hill, with Dunne unable to force home the rebound.
Hill twice headed over before Austin then squandered a golden chance to break the deadlock after beating the offside trap to collect Wright-Phillips' ball into the box with 13 minutes remaining.
Unmarked and with only Loach to beat, Austin opted for power and sent a ferocious shot against the underside of the bar when it seemed easier to score.
Austin was handed another chance following more good work from Wright-Phillips and was unable to beat the excellent Loach from close range - but this time Hitchcock was on hand to add the decisive touch.