With the referee about to blow for time, Swift's attempted clearance landed at the feet of Steven Masterton and gave the Clyde player the opportunity to become the home hero.
Keeper Barry-John Corr did get a hand to the ball, but it was not enough to prevent it from going into the net.
In a goal-less first session, Clyde found a well-organised Stranraer outfit difficult to break down, and their own defence had problems handling quality deliveries into the box from visiting midfielder Lee Sharp.
There where chances at both ends with Michael Moore sending a header streight at Peter Cherrie, and Corr producing an instinctive stop to frustrate Alex Williams.
The Clyde striker thought that he had scored with his powerful header was only six yards out, but Corr somehow got a hand to it and his defenders completed the clearance.
Sharp missed a great chance for the visitors just after half-time when in a suspiciously offside position he managed to shoot against the keeper when scoring looked much easier.
Before the winner a Stephen O'Donnell shot was saved by keeper Paul Walker's legs at the expense of a corner, and it was Swift's header following the flag-kick which led to the winning goal.