But the Steelmen's hopes of a first win since Halloween turned into another sob story as McGrath tucked home from the spot with 12 minutes remaining.
The point is enough, however, to see Alexander's new side climb above Hamilton and out of the bottom two while Saints fall to eighth, a point behind Kilmarnock.
On Friday, former Scotland defender Alexander insisted there was no need for major surgery despite inheriting a team without a win in their last 10 games.
Striker Eamonn Brophy started on the bench for Saints after completing his move from Kilmarnock on the eve of the game but fellow new recruit Collin Quaner will have to wait until next week to join the Buddies' Covid bubble.
Saints started the brighter and should have netted the opener 12 minutes in when Declan Gallagher's misplaced clearance fell straight to Marcus Fraser, but it was a defender's finish as the right-back dragged wide on the volley.
Jim Goodwin's team had a couple of other half-chances go astray and having failed to capitalise when on top, they were punished at the other end.
And they were warned before the goal came. Tony Watt should have done better than head straight at Jak Alnwick from a superb Liam Polworth cross, while Cole rifled wide of target after his instant control opened up space just outside the box.
But Well did convert on 27 minutes as Saints failed to deal with another menacing Polworth delivery. Connor McCarthy's header barely made it 10 yards before Watt nodded it back into the danger zone, where Cole rammed home.
Stephen O'Donnell's last-ditch tackle on Dylan Connolly just before half-time ensured the Steelmen kept their lead at the break.
But Saints were wasteful again with their finishing just after the hour mark as Fraser's throw-in found Jamie McGrath in the box. The Irishman span well before cutting back for Ethan Erhahon but the midfielder's low shot squirmed wide to leave Goodwin cursing on the touchline.
However, the Saints boss' prayers were answered with 12 minutes left as Bobby Madden pointed to the spot when Erwin was felled in the box after a tangle with Gallagher and Bevis Mugabe.
Brophy was on the pitch by this point but it was McGrath who to stepped up to slot down the middle.
Kelly did get hands to Ilkay Durmus' free-kick as he prevented Saints grabbing the winner. Well had their own set-piece chance moments later but Alnwick's save was even more impressive as he clawed O'Hara's set-piece away from the top corner.