QPR had looked comfortable in the first-half but Olivier Giroud opened the scoring on 64 minutes and Sanchez doubled the lead five minutes later as Arsenal, who remain in third, extended their winning run to four league games.
Charlie Austin grabbed the goal his performance deserved late on but it was just a consolation.
With Aston Villa winning on Tuesday, QPR are now three points from safety, albeit with a game in hand.
Early signs suggested that this would be an uncomfortable evening for the visitors with QPR intent on targeting Arsenal’s new signing Gabriel with the muscle of Bobby Zamora.
The defender struggled to deal with Zamora’s physical play and was replaced by Laurent Koscielny midway through the half after taking more than few knocks from the QPR frontman.
The hosts failed to profit from Zamora’s dominant play in the final third, though, with Matt Phillips coming closest with a cross-shot that was palmed away by David Ospina.
Austin also went close with a rasping 25-yard drive that skimmed the post.
Arsenal struggled to get their midfield creators into the game in the first 45 minutes but came out with fresh impetus after the break.
Mesut Ozil’s appeals for a penalty on 47 minutes were rejected when he appeared to be tugged back in the six-yard box by Karl Henry while trying to latch onto a Tomas Rosicky cut-back.
But Arsenal continued to press and got the goal their pressure deserved through Giroud, who has now scored five goals in his last five games.
Young QPR full-back Darnell Furlong was left exposed on the right and Sanchez ran at him with intent before shifting the ball to Kieran Gibbs, whose shot was blocked by Clint Hill but followed up from three yards by the Frenchman.
Furlong was struggling to cope with Sanchez’s direct running and the Chilean got his goal – his first since the 3-0 victory over Stoke in January – on 69 minutes.
He should have scored two minutes earlier but though he fluffed his lines when through on goal after a Steven Caulker error, he was foot perfect when driving to the by-line, cutting back onto his right foot and beating Rob Green at his near post with a low drive.
That five-minute blitz knocked the stuffing out of QPR and Ozil nearly made it 3-0 on 76 minutes, firing a deflected shot onto the post via the outstretched hand of Green.
Austin gave his team hope with eight minutes to play, finding space inside the area between Per Mertesacker and Gibbs before smashing a fine effort into the top corner.
QPR threw defender Caulker into the attack in the final stages but Arsenal stood strong to continue their fine run of results in 2015 – only Liverpool have taken more points this year.
Next up, Manchester United in the FA Cup.
QPR: Green (6), Furlong (5), Caulker (5), Onuoha (6), Suk Young (6), Phillips (6), Henry (6), Sandro (6), Hoilett (6), Zamora (7), Austin (7).
QPR subs: Hill (6), Isla (6), Krancjar (6)
Arsenal: Ospina (7), Bellerin (8), Mertesacker (6), Gabriel (5), Gibbs (6), Coquelin (7), Cazorla (6), Rosicky (6), Ozil (6), Alexis (8), Giroud (7)
Arsenal subs: Ramsey (6), Welbeck (6), Koscielny (7)
Man of the Match: Alexis Sanchez