The Dark Blues, who received a late summons to the top flight following Rangers' demotion last summer, are now 10 points behind the Paisley side with four games remaining.
Dundee hit the post through Kevin McBride and went ahead in the 40th minute when John Baird's flighted pass caught Jim Goodwin flat-footed and Jim McAlister (40) had time to pick his spot before cracking a shot beyond Craig Samson from 14 yards.
Steven Thompson came close to an instant riposte as he volleyed wide from 18 yards but the St Mirren striker's aim was true in the 52nd minute as he met Gary Teale's corner to head home his 16th goal of the season.
Veteran keeper Steve Simonsen produced a fine instinctive save in the visitors' goal to deny substitute John McGinn.
Goodwin, who had been treading a fine line since picking up a first-half yellow card, earned his marching orders when he tangled with Finnigan in the 65th minute.
Gary Harkins fired wide as the Dark Blues looked to make their numerical advantage count and they finally reaped their reward when Finnigan's header from Gary Harkins cross (81) was deemed to have crossed the line after hitting the crossbar.