Both sides saw long-range efforts drift over the bar in the early stages, and Peter Cherrie in the Clyde goal had no difficulty with a Stewart Kean header from a David van Zanten cross.
Clyde's Tom Brighton left Alan Reid and Simon Lappin for dead, but his low cross was cleared.
Out of the blue Mirren scored. Simon Lappin made ground on the left flank and his cross was met by Kean, whose effort was blocked. Marc Corcoran swooped to send a low drive past Cherrie from eight yards.
Corcoran was foiled by Cherrie as he raced on to a superb Charlie Adam pass, but referee Richmond - inconsistent and fussy - waved play on as the striker lay prostrate.
Four minutes from the break Kevin McGowne tripped Adam in the box after the mid-fielder's surging break and Kean smashed in the resultant spot-kick.
Two minutes into the second half Clyde reduced the arrears when Brighton collected a through ball and clipped it past Chris Smith, with the keeper looking hesistant.
Clyde attacked relentlessly and both Alex Williams and Gary Arbuckle missed two chances each.
Kean had a fierce shot brilliantly saved by Cherrie and Adam curled over a late effort, but Saints looked nervous up to the end as the visitors pressed, even though Clyde's Stephen O'Donnell was dismissed for his second yellow card.