Griezmann made light of a dismal recent record from the spot to send Kasper Schmeichel the wrong way in the 28th minute but Marc Albrighton's clear foul on the France star was made just shy of the box.
Diego Simeone's side are chasing their third final appearance in four seasons and, having overcome Barcelona and Bayern Munich over two legs last term, they did not always appear comfortable in the role of favourites.
Atletico have not conceded a goal in a home knockout tie in Europe since March 2014 and Leicester failed to register a shot on target as they rarely looked like breaking that run.
But Atletico were unable to turn their dominance into a more convincing win and Leicester, who overturned a first-leg deficit to knock out Sevilla in the last-16, will still have designs on another LaLiga scalp.
Atletico pushed forward from kick-off, with Saul Niguez having a shot blocked, but the sight of Jamie Vardy tearing after Shinji Okazaki's punt and firing across the face of goal demonstrated Leicester's most likely route to an upset.
Koke powered a rising 25-yard strike against the left-hand post as Schmeichel grasped at thin air, while Yannick Carrasco failed to punish Robert Huth for making a mess of a headed clearance in the ninth minute and slashed his back-post volley wide.
After 15 minutes, Leicester had registered a mere 10 touches inside the Atletico half but the visitors belatedly settled into a period of possession and Okazaki was unable to readjust adequately to turn home Albrighton's dangerous low cross.
The sense that Simeone's counter-attacking masters were ushering Leicester onto them was heightened by the manner in which they broke to prompt the opening goal.
Griezmann tore towards goal from the left with rapid menace and a panicked Albrighton brought him down, with referee Jonas Eriksson pointing to the spot, though replays confirmed the contact came outside the area.
Schmeichel saved a penalty in each leg against Sevilla but Griezmann, who began a sequence of three consecutive misses amid Champions League final heartache against Real Madrid last term, coolly stroked his kick into the left corner.
Atleti were unable to add to their lead before the break, when Craig Shakespeare sent on Andy King in place of Okazaki to bolster Leicester's over-run midfield.
The Wales international's introduction did little to alter the pattern of the play, although Huth - later booked for a foul on Griezmann to rule him out of next week's second leg - had a shot blocked when Atletico failed to fully clear Christian Fuchs' long throw.
Fernando Torres should have doubled Atleti's lead after the hour when he gathered Koke's probing pass and scooted around Huth, only to lose his footing in comic fashion.
Atletico substitute Angel Correa passed up a late chance with the goal gaping, by which stage Leicester deserved credit for drawing the sting from Atletico and Shakespeare's men will still feel the semi-finals are tangible ahead of next week's return at the King Power Stadium.
KEY OPTA STATS
- Atlético Madrid have won 17 of their 22 Champions League home games under Diego Simeone (D4 L1), with the Spanish club unbeaten in the knockout stages (W6 D4).
- Leicester have lost on each of their three European trips to Madrid, with Atlético still unbeaten at home against English sides (W6 D5).
- The Madrid club have progressed in six of their last eight European cup ties against English opposition.
- Atlético Madrid have also kept a clean sheet in 16 of their last 18 Champions League games at the Calderón.
- In fact, keeper Jan Oblak has kept clean sheets in each of his six Champions League knockout games at home for Atlético Madrid.
- Antoine Griezmann has been directly involved in 10 goals in his last 9 Champions League appearances at the Calderón (8 goals, 2 assists).
- Four of Atlético’s last five Champions League goals at home have been scored by Griezmann (Kevin Gameiro the other).