The £11 million (€13.7m) summer acquisition headed home his first goal for the Hammers after just three minutes and began what was mostly an uncomfortable evening for the home support at Loftus Road.
QPR looked devoid of ideas and were on the back foot throughout a first half in which Ricardo Vaz Te acrobatically doubled the visitors’ advantage.
Adel Taarabt clawed a goal back for QPR with a superb long-range effort moments after his second-half introduction, but any comeback was thwarted when fellow substitute Samba Diakite was sent off after collecting two bookings in the 20 minutes he was on the field.
Referee Mark Clattenburg handed out 11 bookings in all and eight of those went to players in claret and blue, who will be fined for indiscipline but held out for three valuable Premier League points at the expense of rock-bottom QPR.
West Ham captain Kevin Nolan was understandably delighted with a first away league victory of the season.
Nolan said: “It was absolutely vital. Delighted with the win today.
“They made us work for the win in the second half. It was a fantastic first-half performance, the second half I thought we dug in and we could have won it by more in the end.”
Nolan also praised his side for the way they fought to hold off QPR’s attempted comeback after the break.
“We knew at some point they had to make changes, we had to be ready for that and I thought we dealt with it well,” he said.
“It was a great goal (from Taarabt) but I thought after that we responded well, we dug in, they had the man sent off which helped us, but I thought overall we were deserving of the win.”
Coming into the game, the surprise team news was that Andy Carroll was fit enough to be included amongst the substitutes after missing the past four matches with a hamstring injury.
And Hammers fans had a goal to cheer after just three minutes.
Kevin Nolan linked up well with Vaz Te down the left but struck a poor scooped effort that was going harmlessly wide until Jarvis snuck in to nod home past Julio Cesar.
It was a bolt out of the blue in what was an unspectacular opening, although the Hammers came close to a second in the 11th minute when Mohamed Diame’s 25-yard thunderbolt deflected off Ryan Nelsen and flew just over.
Winston Reid and Ireland’s Joey O’Brien were forced off injured for West Ham but the Hammers doubled their lead through Vaz Te just before the half-time whistle.
Mark Noble swung in a free-kick and the resulting penalty-box melee was cleared only as far as Tomkins, whose superb first-time cross was met by an equally impressive acrobatic effort from Vaz Te.
West Ham started the second half the brighter and it took until the 55th-minute introductions of Taarabt and Diakite to change course.
The former was welcomed with particularly loud cheers and showed why within two minutes, making space to curl home an exquisite 25-yard effort into the top right-hand corner.
But Taarabt was booked for lifting his shirt and was soon joined in Clattenburg’s notebook by fellow replacement Diakite who saw red for a second booking 16 minutes from the end.
QPR: Julio Cesar, Hill, Nelsen, Mbia, Onuoha (Hoilett 84), Wright-Phillips (Taarabt 56), Granero, Faurlin, Park (Diakite 55), Zamora, Cisse.
W HAM: Jaaskelainen, Demel, Reid (Tomkins 23), Collins, O’Brien (McCartney 35), Noble, Diame, Vaz Te, Nolan, Jarvis, Cole (Carroll 72).
Referee: Mark Clattenburg.