A deflected strike from Darron Gibson - Everton's 50th goal in the Premier League this season - and a smart finish from Victor Anichebe five minutes either side of half-time secured a victory the hosts were just not made to work hard enough for.
With five matches remaining the west Londoners are seven points from safety and they simply have to win their next two games against Stoke and Reading to even stand a chance of surviving.
Junior Hoilett's curling effort in the fourth minute after Andros Townsend's 40-yard run at the centre of the defence was bettered in the first half only by Loic Remy's powerful effort which Tim Howard parried for a corner.
Everton were also not their usual busy selves, although the tempo of the match may have changed had referee Lee Mason felt Clint Hill's challenge on Seamus Coleman warranted a penalty. He did not and it was to the hosts' detriment as they toiled for 40 minutes without really opening up a side second-bottom in the table.
Anichebe was inches away from connecting with Leighton Baines' cross while Phil Jagielka failed to get his head onto the England left-back's inswinging corner.
Baines again provided an opportunity when he cut back a right-sided free-kick to Gibson outside the box. The Republic of Ireland international rolled a pass intended for Marouane Fellaini, Everton's top scorer returning after suspension, only for Sylvain Distin to get there first but bundle his shot hopelessly wide from eight yards out.
When the home side eventually took the lead it was through a stroke of luck as Gibson's long-range strike took a significant deflection off Hill to leave goalkeeper Julio Cesar helpless.
If Hoops boss Harry Redknapp expected a response from his players after the break he would have been disappointed as it was Everton who came out more determined, but with not the slightest attacking intent or invention visible it was no surprise when QPR conceded again. Distin headed goalwards and Anichebe neatly hooked home a close-range shot in the 56th minute.
The introduction of the mercurial Adel Taarabt injected some life into the visitors lacklustre approach but even then they tested Howard just once in the final half-hour when full-back Nedum Onuoha forced the American into a save.
Source: PA
Source: PA