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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.