The £11million 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 from the outset and were on the backfoot 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.
Coming into the game, the surprise team news was that Andy Carroll was deemed fit enough to be included amongst the substitutes after missing the past four matches with a hamstring injury.
Former West Ham goalkeeper Rob Green was on the opposing bench and was taunted from the start by the vocal visiting support, having lost his place in the QPR team just four games after his summer move from Upton Park.
The England international was reminded of his decision throughout the match by 1,681 away supporters, who 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.
Joey O'Brien reacted quickly to stop James Collins putting into his own goal from an Esteban Granero free-kick that saw an unfortunate collision between Winston Reid and his own goalkeeper.
The New Zealand international required nearly four minutes of treatment after appearing to be knocked out by Jussi Jaaskelainen and saw James Tomkins come on in his place.
The break in play did not change the game's momentum as only Stephane Mbia's quick reactions prevented Diame getting a head on a right-wing Nolan cross.
It was not all positive for West Ham, though, as O'Brien became their second casualty of the first half and had to be replaced by George McCartney 11 minutes from half-time.
The injuries were the only blot on a fine opening 45 minutes for West Ham, which got better as Vaz Te doubled the visitors' advantage moments later.
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.
"Greeno, what's the score?" rung around Loftus Road following the goal and would have gone up a notch had a Carlton Cole header or Noble's deflected long-range strike found the target.
The latter and Diame were both booked heading into the break from which West Ham returned on top.
Cesar did well to deny Diame three minutes after the restart 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.
Taarabt was booked for lifting his shirt and was soon joined in Clattenburg's notebook by fellow replacement Diakite.
West Ham's McCartney was the next, before Jaaskelainen was reprimanded shortly after denying Granero with an excellent stop.
Cole and Nolan both received yellow cards as the match became a bit of a farce and Allardyce soon brought on Carroll for the former.
Jaaskelainen spilled a driven Cisse shot at the near post as QPR pushed for an equaliser, before the hosts' task was made all the more difficult as Diakite received a second yellow a late lunge on Diame.
Vaz Te had an excellent strike palmed onto the bar by Cesar after Collins became the seventh West Ham player to be booked.
Cisse and Alejandro Faurlin were both denied by Jaaskelainen as QPR pushed for a leveller that never came as West Ham held out for victory, although not before Tomkins became the final player booked.