Dundee's long-anticipated relegation remains on hold thanks to a contentious, late winning goal from Carl Finnigan against 10-man St Mirren.
Jim McAlister shot the visitors into a half-time lead but Steven Thompson levelled with a header after the break.
St Mirren skipper Jim Goodwin was then shown a second yellow card.
Finnigan's header struck the bar and was ruled to have crossed the line to reduce the gap between the sides to 10 points with four matches remaining.
The drop for Dundee is almost inevitable but they are certainly going down with a fight.
The bottom side have lost just one of their seven league matches since John Brown arrived as interim manager.
The spirit displayed by Dundee must be applauded but Goodwin's foolish dismissal played a large part in their success, breathing fresh life into their cause during a period in which the Buddies had the upper hand.
There was an early scare for the home side when Baird raced past Goodwin but goalkeeper Craig Samson got down to make a brave save at the striker's feet.
Dundee keeper Steve Simonsen then had to look lively to tip over a delivery from Gary Teale that was caught by the stiff breeze swirling around the stadium.
Lewis Toshney thundered a shot wide and a Kyle Benedictus free-kick was comfortably saved before Dundee came within inches of the lead.
Baird did well to keep a high ball in play and fed McAlister, who had a strike on the turn headed clear by Goodwin. It fell perfectly for Kevin McBride to crack a wonderful volley against the junction of post and crossbar from 25 yards.
Thompson was claiming for a penalty when his header struck a raised arm in blue and the St Mirren striker was still complaining as he took a wild fresh-air swipe when the ball was fed back into the penalty box.
Dundee were showing a greater appetite for the fight and were rewarded four minutes before the interval when Baird lifted a cute pass over the static Goodwin and McAlister had time to let the ball bounce before lashing a powerful shot past Samson from 12 yards.
Thompson came close to an equaliser moments later when he caught a headed clearance flush but his volley fizzed narrowly wide.
Thompson's mood improved early in the second half when he rose highest at the back post to thump in a header from Teale's corner.
St Mirren substitute John McGinn thought he had scored with a close range header but Simonsen made a brilliant save to claw the ball away and the loose ball was blazed over the crossbar.
The home side were in the ascendancy but lost their momentum when Goodwin was sent off.
The Irishman was booked for a crude first-half tackle and earned his second caution for clattering into Finnigan, just a few minutes after a bad-tempered tangle near the corner flag with the same opponent.
Buoyed by their numerical advantage, Dundee pressed forward and Finnigan appeared to be impeded in a busy penalty area before Gary Harkins drilled a low shot into the side-netting.
The home side packed men behind the ball but Teale was a threat on the break and Simonsen did well to smother a strike from the veteran winger.
The small band of Dundee fans may have thought their luck was out when Brian Easton thundered a shot against the post but Finnigan was to grab a dramatic late winner.
The substitute threw himself at a Harkins cross and glanced a header against the underside of the crossbar. The ball bounced down sharply but the assistant referee immediately signalled a goal and the Dark Blues live to fight on for another week at least.