Griezmann was fouled by Leicester's Marc Albrighton on the edge of the box and referee Jonas Eriksson deemed the action to have continued into the penalty area.
Griezmann sent Leicester goalkeeper Kasper Schmeichel the wrong way from the spot after 28 minutes although the host could not add to their advantage in the remainder of the encounter and Fernando Torres missed a good chance to double the lead.
The result left the tie in the balance ahead of the second leg in England on April 18, as Leicester bid to maintain a fairytale run in their first season in Europe's elite club competition.
Atletico dominated from the start, with midfielder Koke slamming a long-range strike against the post in the fourth minute.
Griezmann and Saul Niguez both had drives from distance which flew narrowly off-target as Diego Simeone’s side pushed for the opener.
It eventually came after Leicester’s first real attack of the game, with Atletico surging forward on a lightning counter, which ended when Albrighton brought down Griezmann on the edge of the area.
The Leicester winger’s first contact with the French forward took place outside the box although referee Eriksson pointed to the spot. Griezmann dispatched his effort coolly to notch his 24th goal of the season.
Leicester stayed tight at the back and although Atletico dominated possession, they could not create any more clear-cut chances in the first half.
After the break Torres escaped his marker in the area but slipped as he was poised to strike, with the ball bobbling towards the corner flag and going out for a throw.
As Atletico pushed in vain for a second goal, Leicester's hopes of overturning the deficit in the second leg were hit when Robert Huth was booked for a foul on the effervescent Griezmann, ruling the defender out of the return encounter.