The 20-year-old St Johnstone striker has started the season on fire for his club side with seven goals and carried his impressive form onto the international stage with Scotland's second at St Mirren Park.
Home skipper Stuart Armstrong had handed Billy Stark's side the lead after half an hour with a tidy finish but the Slovakians did give them cause for concern late on after substitute Thomas Malec halved the deficit with a close-range finish.
The victory lifts Scotland above their opponents and into second place in Group Three, with Georgia next up in Tbilisi on Tuesday.
Stark's youngsters were desperate to post a respectable result having shipped 10 goals in successive hammerings by England and Holland.
But they were up against a side who arrived in Scotland top of a qualifying group that also includes the Dutch and Luxembourg following victories in both of their opening qualifiers.
However, Scotland did not let their opponents' soaring confidence deter them from getting forward and after overcoming a nervy opening, found two goals to settle themselves down.
First on 31 minutes, Celtic defender Stuart Findlay sent Clark Robertson down the left and the Aberdeen youngster's cut-back was perfect for Armstrong to steer into the corner of the net.
And a second came soon after as May gathered the ball in the inside left channel before shrugging off Norbert Gyomber as he strode into the box. The angle looked improbable but that did not stop the confident Perth striker from lashing a left-foot drive over Patrick Legiand into the roof of the goal.
Yet the Scots were lucky not to end the half a man down after Fraser Fyvie's dangerous two-footed lunge on Jakub Paur was punished only by a yellow card from Croatian referee Damir Batinic.
May was denied a second on the stoke of half-time as Branislav Nidaj hacked off the line after the St Johnstone man had strode past the grounded Legiand.
Scotland keeper Jordan Archer had to block at his near post a minute into the second period as Malec almost made an immediate impact moments after coming on, while Fyvie failed to make the most of a swift breakaway 30 seconds later despite being through on the goalkeeper.
The Dark Blues continued to throw bodies forward but a killer third goal did not arrive.
A May header on 54 minutes came back off the cross bar following a teasing delivery from Celtic's Dylan McGeouch, while his Parkhead team-mate Callum McGregor - currently on loan at Notts County - saw a lot shot blocked by Legiand after May put him through.
But with Scotland wasteful with their finishing, Malec increased the tension in the home ranks when he tapped home from Jaroslav Mihalik's low centre with 22 minutes left.
Fyvie that had to smash clear from under his own crossbar after Malec beat Archer to a corner as the visitors piled on the pressure but Scotland held out to pick up their second win of the qualifying campaign.
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