The France striker was fouled outside the box by Marc Albrighton but referee Jonas Erikssson pointed to the spot - yet the Foxes' European hopes remain alive.
It was deja vu for Leicester after another dubious penalty at the Vicente Calderon helped knock them out of the UEFA Cup in 1997.
They will also have to overturn a deficit at the King Power Stadium on Tuesday without Robert Huth, who is suspended after collecting his third booking of the campaign.
But the Foxes, who failed to create a meaningful chance, will be encouraged they are in the tie and still have a chance of booking a stunning semi-final spot.
They defended well and were vigilant although Atletico's one-paced second-half performance played into their hands.
Atletico dominated from the off and Koke smacked the outside of the post from 25 yards after five minutes before Yannick Carrasco volleyed wide after Huth's miscued header.
Similar to the last 16 first leg against Sevilla, Leicester were pinned back by their Spanish hosts as Atletico marauded forward with growing menace.
The Foxes were sloppy and failed to help themselves as they gifted possession back to Atletico with striker Griezmann firing over.
There was little early respite as the frantic Leicester defence held firm but slowly the Foxes began to make some ground.
Wilfred Ndidi had been vital in stemming the Atletico attack but the visitors were undone in controversial style after 27 minutes.
They had looked more comfortable but were caught by a lightning break when Griezmann burst clear on the left. The striker went to sprint into the area but he was clipped by Albrighton and referee awarded a penalty - despite Griezmann being clearly outside the area.
There were echoes of 1997 - when Delfi Geli went down softly under Steve Guppy's challenge to win a penalty in their UEFA Cup meeting - but Griezmann ignored the debate to beat Kasper Schmeichel.
Atletico deserved the lead, if not the opening, and Koke went close to making it 2-0 when he lashed wide from 25 yards.
Leicester held firm ahead of the break, with Albrighton firing wide, and Andy King replaced Shinji Okazaki at the interval with boss Craig Shakespeare searching for a way back.
Atletico played a deep line to nullify Jamie Vardy's threat after boss Diego Simeone pinpointed him as the main danger before the game.
The striker continued to find it tough but the Foxes almost found a way through when Huth's close range effort was blocked after 54 minutes. But the defender was booked minutes later when he clipped Koke and will now be banned for the second leg.
Leicester then survived a scare when Huth was beaten by Fernando Torres but the former Liverpool striker slipped as he shot and the ball bounced out for a throw in.
Methodical Madrid continued to try to unpick Leicester and Angel Correa was inches away from connecting with Griezmann's cross with 16 minutes left.
The Foxes, though, remained resolute and still have a chance to keep their European dream alive and clinch an improbable semi-final spot.