Christian Gentner had fired the hosts ahead before half-time of the Europa League last-32 match, but Plet struck in stoppage time to level the scores and give his team the advantage heading into the return match in Belgium next week.
The two teams had differing fortunes in the group stage, with Genk unbeaten while Stuttgart scraped through to the knockout rounds on head-to-head record.
The German side also came into the match on a five-match losing run in the Bundesliga but they registered the first chance after 15 minutes, with Vedad Ibisevic's strike drawing a brilliant reflex save from Genk goalkeeper Laszlo Koteles.
The Belgians responded within minutes on a corner from Benjamin De Ceulaer, which Elyaniv Barda headed narrowly past the left post.
Stuttgart threatened again on the half hour with a right-footed shot from Martin Harnik which was also palmed away by Koteles.
Home goalkeeper Sven Ulreich then made his first real contribution, turning away a right-footed drive from De Ceulaer.
Chances were coming regularly now at both ends of the pitch, but it was Stuttgart who were the first to find the target three minutes before half-time.
Gotuko Sakai's left-footed effort was pushed away by Koteles but Gentner was on hand to smash home the rebound from close range.
Genk tried to pull level before the interval but Ulreich was equal to Jelle Vossen's snap-shot.
Stuttgart looked confident in the early stages of the second half as Georg Niedermaier powered a header over the bar from a few yards out.
Ibisevic also fired over the target from 20 yards as the Bundesliga outfit looked to increase their lead, while his next chance was a header which dropped wide of the right post.
And the Bosnian made it a hat-trick of misses after 72 minutes with a drilled effort which found the safe hands of Koteles.
With chances few and far between in the final 15 minutes, it looked as if Stuttgart would hold the advantage going into the return leg.
But Genk, who had done little of note in the second half, snatched an unexpected draw in stoppage time when Plet headed home Steven Monrose's cross from the left.