It looked like being a frustrating night for the Tartan Army, who twice hit the crossbar in a frenetic second half, but Skrtel diverted the ball past his keeper in the closing stages.
Scotland, knowing that three points were a must if they were to extend their campaign into the final round of matches, started the better and felt that they should have had a penalty eight minutes in.
Kieran Tierney was bundled over in the box by Robert Mak who, although fortunate on this occasion as play was waved on, was not so lucky soon after when he was dismissed for a second bookable offence.
Minutes after being cautioned for a challenge out wide, the Zenit St Petersburg ace - on the scoresheet twice in the reverse fixture - saw a second yellow for a clear dive inside the box.
For all of Scotland's dominance they could not create too many clear-cut openings and Slovakia, more than happy with a point on their travels, sat back and soaked up the pressure.
The visitors had the odd positive moment in the first half, the best of which saw Adam Nemec's shot blocked from the edge of the area as they looked to make the most of counter-attacking opportunities.
Scotland, who had scored in 14 successive games at Hampden heading into this match, next came close through a Griffiths header that Dubravka was able to smother.
Home fans desperately urged their side on in search of a winner, seeing another Griffiths drive helped over the bar and a well-hit strike from substitute Chris Martin cannon back off the frame of the goal.
Griffiths was the next home player to hit the bar, doing so from a 25-yard free kick, while James Morrison was somehow denied by an inspired Dubravka from close range seconds later.
Nearly three-quarters of Scotland's goals this campaign have come in the second half of games, five of those in the final six minutes, and another late goal was required here.
Just when it all looked to be over, Martin again proved to be the late hero as he applied the pressure on Skrtel to turn the ball past his own goalkeeper.
Scotland now head to Ljubljana knowing that victory should be enough to book a best-placed runners-up spot ahead of next month's playoffs.