It didn't take long for St Johnstone to assert themselves in this match, taking the lead in the third minute. World Cup hopeful Jason Scotland displayed individual brilliance in dancing around two defenders before slotting past Barry-John Corr to score his 12th league goal of the campaign.
St Johnstone nearly doubled their advantage in the 14th minute, but Corr was able to block Martin Hardie's header.
In the 21st minute relegation-threatened Stranraer pulled level with Wille Martin slamming the ball home from 10 yards.
Unfortunately the visitors were unable to stay on level terms until half-time as the Saints, inspired by substitute Steven Milne, were looking likely to score.
In the 36th minute Milne worked well to set up Scotland, but his shot was parried wide by Corr. Scotland wasn't to be denied for long, though, and from the resulting corner he forced the ball over the line following a magnificent looping header by strike partner Milne.
Stranraer dragged themselves level again in the 55th minute with midfielder David Hamilton heading powerfully home from a Stephen Payne free-kick to make it 2-2.
St Johnstone were not to be denied, however, and scored the winner in the 74th minute. It was a tremendous flowing move with Stevenson releasing his manager, Owen Coyle, to put the ball on a plate for top scorer Scotland to bundle home.