McKay's 74th-minute header prevented the Blues from marking Caley Thistle's 14th anniversary of their famous shock cup win over Celtic with defeat.
Aaron Doran had put the League Cup finalists ahead five minutes before the break but Martin Grehan levelled from the spot on the stroke of half-time and Jamie Longworth put the minnows ahead after the restart.
However, McKay struck two minutes later to secure a 2-2 draw and a fifth round replay and maintain Caley Thistle's hopes of a cup double.
The Scottish Premiership Highlanders knew they faced a game from early on when Stranraer missed a great chance in the opening moments.
Grant Gallagher fed Sean Winter but, instead of shooting first time, the winger took a touch which allowed visiting goalkeeper Dean Brill to charge down the shot.
Steven Bell also had two shots from distance at Brill's goal as Stevie Aitken's men sought to do a Highland double after sending Ross County tumbling out of the League Cup in August.
Ross Draper should have given Thistle a foothold in the match but he could only divert his free header straight at David Mitchell before Doran brilliantly curled Caley Jags in front, from 25 yards (40).
A quick throw-in by Graeme Shinnie caught Bell unaware and the midfielder missed his header to allowed Doran a free run into the box.
The winger kept his cool and curled his shot into the bottom corner, leaving Mitchell rooted to the spot.
Stranraer would have felt hard done by to go in behind at the break and they were soon handed a lifeline by referee Kevin Clancy.
Grehan was hauled from behind by Shinnie just as he was about to convert a Longworth cross and the official had no hesitation in pointing to the spot.
The striker took the penalty himself and sent Brill the wrong way.
Thistle manager John Hughes wasted no time bringing on Richie Foran for his first game since suffering a shoulder injury against Aberdeen on December 21 in a bid to increase his attacking options.
Despite the extra forward, Thistle still struggled to create attacking opportunities, although Blues skipper Frank McKeown threw himself in front of a McKay shot with Mitchell beaten.
Stranraer got the break all their hard work deserved midway through the half when Longworth pounced on a loose header on the halfway line.
The hitman caught the Caley Thistle defence flat-footed and, despite Brill trying the close him down, he rifled his shot into the net for his 20th goal of the season (72).
The shocked Highlanders regrouped quickly, McKay firing a warning shot from a corner, before the same player nodded in from a Foran cutback.
Both teams brought on fresh legs as they pushed for a winner but both defences held strong to make the teams do it all again a week on Tuesday.