Baird, who had never scored for the Cottagers before, struck twice from outside the area in the opening 10 minutes to lift some of the pressure on his manager Mark Hughes.
The Northern Ireland international was an unlikely goalscorer to seal an unlikely result, as the London outfit recorded their first away win in the league in 27 attempts.
The Potters would have gone into the game feeling confident having already beaten Fulham three times in 2010, but it was the visitors who found themselves ahead after only four minutes.
Simon Davies sent in a cross from the left that was only partially cleared and it fell to Baird, who rifled in a bullet of a shot that Asmir Begovic could not prevent crashing in off the post.
The hosts tried to make a swift response and Robert Huth's shot sailed over, but Andrew Johnson had a chance to make it 2-0 soon after having wriggled through a crowd of players.
Johnson's effort was straight at Begovic, but moments later, Fulham had doubled their advantage, leaving Stoke and their supporters stunned.
Perhaps most surprised of all was the scorer himself, Baird again hitting the back of the net with a long-range strike, this time having been teed up by a free-kick awarded when Jermaine Pennant had brought down Davies.
The shell-shocked Potters attempted to regroup and slowly started to exert some pressure, with Kenwyne Jones narrowly heading over from a corner.
Dean Whitehead then lashed a shot over and protested that he had been impeded by a foul from Davies, but referee Kevin Friend waved the penalty appeals away.
Fulham were almost in again just after the half-hour mark after Begovic's kick went straight to a Fulham shirt, but Dickson Etuhu skewed his cross and the move broke down.
At the other end, Huth drilled a free-kick into the wall and Mark Schwarzer managed to collect the ball after Rory Delap had delivered a dangerous long throw into the box.
Just before the break Johnson was fed by Clint Dempsey but was denied by a last-ditch tackle from Ryan Shawcross, who then saw a header cleared off the line by Danny Murphy.
Stoke boss Tony Pulis, hoping to give his side fresh impetus in attack, brought on Ricardo Fuller for Jon Walters 10 minutes into the second half and the Potters pressed forward.
The home supporters cried out twice in quick succession that a Fulham player had handled in the box, but Friend judged neither Johnson nor Brede Hangeland had been guilty of the offence.
Stoke then earned a free-kick in a useful position which Pennant curled off-target and Jones rose high to meet a Danny Collins cross but could not guide his header the right side of the post.
Tuncay Sanli was next to be introduced to the action in place of Matthew Etherington before the Potters went the closest yet to pulling a goal back.
Fuller laid the ball to Huth, who hit a low effort that Schwarzer dived to block.
Pennant then saw his shot cannon off a defender as the hosts realised it was not their day.
Fulham boss Mark Hughes believes his players may have achieved a "season-defining" result.
"There was real intensity to our focus before and during the game," Hughes said. "We came here with frustration - that was the thing we were all feeling in terms of where we were sat in the league, but that helped our focus and I thought we were immense.
"Everybody on the field of play was exceptional - big performances from every part of the team, and that is what you need. Maybe that has been lacking in away games, that intensity that you need and resolute mindset. So we set the standard today in terms of that and it could be a season-defining win for us."
Hughes agreed with the suggestion that the result was a monkey off the back with regard to Fulham's away record and was delighted to see his players bounce back from their Boxing Day defeat against West Ham, after which speculation had grown that he could soon be facing the axe.
"This week it seems to have been me, next week it will be someone else - that is the nature of the job and we all accept it," Hughes said. "It is not nice when that focus lands at your doorstep, but I know I have a good group of players that, given the rub of the green and when they play as well as they did today, will have no problems this season."
Asked if he had sought assurances about his future from the Fulham board, Hughes said: "No, not at all. I don't feel I need to. I have got great support from (chief executive) Alistair Mackintosh and I spoke to the chairman (Mohamed Al Fayed) last night, who is very supportive.
"They understand where we are and what we need, and the reasons why we find ourselves in the bottom half. We will look to address them, but in the meantime, the guys I have got playing for me are doing their best, and today was excellent."
Hughes did not receive a handshake after the game from his opposite number Tony Pulis, something the 47-year-old believed was a reaction to how he had conducted himself at the end of the Carling Cup third round tie between Stoke and Fulham in September.
"He is probably getting his own back, because I did a similar thing last time in the League Cup because I was a little bit aggrieved one of his players (Andy Wilkinson) had taken out Moussa Dembele when it was unnecessary," Hughes said.
Pulis confirmed that to be the case.
"He has done it when we played in the League Cup and now I have done it back," Pulis said. "It's two Welshmen with a bit of competition. I certainly won't lose any sleep over it and I'm sure he won't."
Stoke had two penalty appeals for handball turned down in the second half, although Pulis seemed more concerned with a tackle Brede Hangeland made on Rory Delap.
"The one situation that did disappoint me was the incident with Hangeland and Rory," said Pulis. "It is a poor challenge and for the referee not to give a free-kick when he is standing two yards away from it was desperately disappointing.
"He might not book him, but he didn't even give a free-kick and he has really caught Rory. I don't know about the penalties, I haven't seen them again yet."
Fulham captain Danny Murphy mentioned Stoke specifically when talking about dangerous tackling earlier in the season, which prompted an angry rebuttal from Pulis at the time.
Murphy was booed by Potters fans whenever he touched the ball, but Pulis would not be drawn on the matter any further.
"What Danny Murphy said, he said," Pulis said. "We are all entitled to our opinion - I have said my bit, Danny has said his and we move on."
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