But Gordon Strachan's side refused to lie down and got the winner with two minutes left on the clock.
Former Liverpool defender Martin Skrtel turned a low cross from Ikechi Anya past his own keeper to help Scotland move up to second place in Group F.
Victory over Slovenia on Sunday in their final group game will secure a playoff place for Scotland as they look to qualify for the showpiece event in Russia next year.
After Dubravka had made a superb save to keep out a close-range header from Christophe Berra early in the game, the home side were given a major lift on 23 minutes when Slovakia were reduced to 10 men.
Robert Mak was shown his second yellow card after attempting to fool the referee when he went down inside the box despite not being touched by Scotland keeper Craig Gordon.
Scotland threw everything at the Slovakia goal as they attempted to find the breakthrough, but Dubravka came to Slovakia's rescue again on 61 minutes when he made a fine save to keep out a fierce shot from Scotland forward Leigh Griffiths.
Scotland supporters must have feared it was not going to be their night after they hit the woodwork twice in the space of five minutes.
Chris Martin picked up the loose ball on the edge of the penalty area on 68 minutes, but his powerful effort cannoned back off the bar.
After scoring two free kicks against England, Griffiths then tried his luck from a similar range but his curling effort struck the underside of the bar.
But spurred on by a passionate home crowd, Scotland refused to give in and got their just rewards on 88 minutes when substitute Ikechi Anya raced into the box and his low cross was turned past his own keeper by Skrtel.
(Reporting by Marc Isaacs, Editing by Ed Osmond)
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