Goals from Robbie Deas, Findlay Curtis and Joe Hugill lifted Kilmarnock up to 10th and dumped St. Mirren, who lost 1-0 at Dundee, into the relegation play-off place.
Livingston kept themselves alive with a 2-2 draw against Aberdeen on Friday night but have realistically known their fate for quite some time.
After a positive start to the campaign on their return from the Championship, Livie went on a 31-game winless run until last weekend's win at St. Mirren.
Dundee's victory opened up a six-point cushion over the Buddies with three games to go in the battle to avoid the play-off.
A three-way fight for the title resumes on Sunday when Celtic travel to Hibernian hoping to move level on points with leaders Hearts.
The Jambos host third-placed Rangers on Monday as they aim to move a step closer to a first top-flight title in 66 years.