The Turkey international scored the 57th-minute winner to set Leverkusen, who are struggling for form in the Bundesliga, on the way to their first ever win at this stage of the competition.
"This was a very, very good game from us," goalkeeper Bernd Leno, who twice rescued his team in the first half, told reporters.
"We showed a reaction to the criticism of the past week. We had some bad games but tonight we bounced back."
For Atletico, who were left with 10 men when Tiago was sent off following a second booking in the 76th minute, it was only their second loss in a Champions League away game since September 2013.
The Spanish champions will also be missing key defender Diego Godin for the return leg after the Uruguayan was booked and will be suspended in Madrid.
"They play as they do and they do it well," said Atletico coach Diego Simeone.
"We each had a clear chance and they took theirs and we didn't. It could have been a worse result. But I am confident for the return leg."
Leverkusen had hoped the European stage would lift their game after a rocky start to the year sent them down to sixth in the Bundesliga.
They did not have to wait long for a chance with Atletico striker Mario Mandzukic clearing an Emir Spahic effort on the line in the 12th minute.
The Bosnian defender came even closer in the 26th when his thundering drive bounced off the crossbar.
The Spaniards needed almost 40 minutes to find their feet and responded with a chance of their own, Leno clearing a cross at the last moment with Antoine Griezman ready to head in.
Leno saved the hosts with a brilliant reflex effort on the stroke of halftime, palming away a powerful volley from Tiago.
In a very physical encounter in which coaches Simeone and Roger Schmidt had a heated argument on the sidelines, Leverkusen grabbed a deserved lead through Calhanoglu.
Karim Bellarabi drew three players on to him before cleverly flicking the ball on for Calhanoglu who lashed the ball into the roof of the net.
Atletico, who will be without Godin and Tiago in the return leg, were toothless even when Simeone brought on striker Fernando Torres to form a three-pronged attack for much of the second half.
(Reporting by Karolos Grohmann, editing by Ed Osmond)
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