In a game of very few chances, Griezmann proved decisive as it was he who also won the penalty with a lung-busting run from well inside his own half before being brought down by Marc Albrighton midway through the first half yesterday.
The French international put his previous penalty demons behind him to convert from the spot and put Atletico in the driving seat for a third semi-final in four seasons.
However, Leicester still have reason to believe their dream run on their first appearance in the Champions League can continue when the sides meet again on April 18.
"I think we were hard done by," said Shakespeare.
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Atletico coach Diego Simeone lamented his side's inability to kill the tie, but lauded Leicester's defensive effort.
"Maybe we lacked a clinical finish," said the Argentine.
"After the 1-0 Leicester defended very well, sat deep, waited for a counter-attack and we couldn't create more chances.
"They bet everything on the second leg."
However, the 3,000 travelling supporters high in the rafters of the Calderon were largely silenced by Atletico's dominance of the match.
Unusually for Simeone's men they were in almost complete control of possession, but largely limited to shots from long range.
Koke came closest from distance when he drilled an effort off the outside of the post after just five minutes.
Yannick Carrasco then slashed wildly at a volley when well-placed at the back post.
- Griezmann flattened -
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Leicester had been enjoying their best spell of the half when suddenly they were caught short-handed at the back on 27 minutes.
Griezmann collected the ball outside his own area and ran to just inside the Leicester box before being crudely barged over by Albrighton.
Schmeichel had saved a penalty in each of Leicester's two legs against Sevilla in the last 16 and Griezmann had missed his last four for Atletico, including in last season's Champions League final.
Both sides seemed content to settle the tie at the King Power in six days' time with Atletico holding a 1-0 advantage after the break.
Leicester were without captain Wes Morgan due to a back injury and his return to health will be even more important for the second leg as Robert Huth will be suspended after being booked for chopping down Griezmann.
Griezmann was the standout performer and nearly added an assist to the only goal but Angel Correa couldn't quite stretch to reach his dangerous low cross 17 minutes from time.
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