Shane Long's fifth league goal of the season rescued a point for West Brom as QPR were denied only their second home win of the season.
Heidar Helguson gave the hosts the lead by heading in Joey Barton's cross.
Shaun Wright-Phillips thought he had doubled the lead but his effort was disallowed for offside.
Albion felt they should have had a penalty when Danny Gabbidon challenged James Morrison, before Long converted substitute Peter Odemwingie's cross.
It was not the birthday present QPR boss Neil Warnock wanted.
Warnock, who celebrated his 63rd birthday on Thursday, had seen his side lose three of their previous four games.
Yet Barton was inspirational on his return from suspension with the QPR captain turning provider for the first goal midway through the first half during a busy midfield performance.
Barton pounced on a loose ball after Jamie Mackie had lost possession attempting to thread his way through the Albion defence, and the former Manchester City player delivered an inch-perfect cross for Helguson to head home from six yards.
It was the fourth successive home game that the Icelandic striker had scored and it left West Brom, who had managed only 12 goals in their previous 13 league games, with an uphill task to force their way back.
Wright-Phillips thought he had doubled the lead four minutes later with a terrific finish into the top corner from an angle.
But the former Chelsea winger's effort was ruled out for offside although television replays showed the goal should have stood.
West Brom were living dangerously and were missing the creativity of Zoltan Gera,
Jerome Thomas and James Morrison were both guilty of wasting good opportunities, but the visitors failed to seriously test QPR's third-choice goalkeeper Radek Cerny in the first half.
At the other end, Ben Foster was fortunate to escape unpunished after a header by Wright-Phillips drifted narrowly wide of the target, with the West Brom keeper well off his line.
The visitors felt aggrieved when referee turned down their appeals for a penalty when Morrison tumbled in the area under a challenge from former Albion defender Gabbidon.
Yet with time running out and QPR seemingly in control, West Brom equalised with a breakaway goal after Long scored from inside the six-yard area following a fine cross by substitute Odemwingie.
QPR manager Neil Warnock said: "I'm disappointed with the linesman. You expect them to get it right when they're in line.
"You wonder how they feel when they see mistakes like that later on television. Whether he got conned by the Tony Adams syndrome, when you run out and appeal, I'm not sure.
"But we had loads of chances and have to learn to put them away. It's all right me blaming the linesman, but it wasn't him who missed all the chances."
West Brom manager Roy Hodgson said: "We worked hard for the result but we made life difficult for ourselves. It was just a case of plugging away and making sure we carried on believing in ourselves.
"It was nice to see Odemwingie back. He got better and better the longer he was out there."