Paul Hartley side are some 16 points adrift of runaway leaders Queen of the South but remain poised to take advantage of any slip-up.
Graeme Holmes netted twice in the second half – and also struck the post early in the game – to assure the home side of the points.
A strike from Ryan McCord was all they had to show for their first-half play and that was wiped out not long into the second half by Grant Gallagher.
Mitch Megginson headed Alloa back in front before Holmes ensured there was no way back for Stranraer.
East Fife and Ayr United served up a warming six goals in a 3-3 draw at New Bayview.
Ayr had to rely on a goal from Cammie MacDonald with just five minutes left to salvage a point after it had looked good for them early on when they led 2-0 through Michael Moffat's 12th goal of the season and another from Kyle McAusland.
East Fife scored three times before the conclusion of the first half – a Paul McManus double adding to David White's first – to turn the tables but MacDonald had the final say.
Arbroath beat Forfar Athletic 3-1 at Gayfield with Graham Bayne, their new signing, proving influential.
He set up the first goal for Lee Sibanda with an early header and then scored himself shortly afterwards, with another header.
Bayne scored his second from close range before Forfar pulled a goal back through an Iain Campbell penalty.
Any hopes they had of taking anything effectively ended when Willie Robertson was sent off with 13 minutes of the match remaining.