Vincenzo Montella's side booked their spot in the knockout stage of Euro 2024 on Wednesday night courtesy of a 2-1 success over Czech Republic, which saw them finish second in Group F.
Calhanoglu sent Turkey ahead in the 50th minute of the clash with a rasping strike, but the 30-year-old, who plays his club football for Inter Milan, will be suspended against Austria on Tuesday, having been booked for remonstrating with the referee after the Czech Republic goal.
Montella paid tribute to the captain for his "extraordinary goal" in what was an ill-tempered affair, with 16 yellow and two red cards being issued by the referee.
"He managed to score an extraordinary goal. If you look at the way he took the shot, there are few players in the world that can take a shot like he does," the Turkey head coach told reporters.
Calhanoglu will be absent against Austria
"And sending the ball to the far corner using the outside of your foot, he's an extraordinary player. And he's an extraordinary man.
"These 18 bookings show how intense the match was. Calhanoglu was just asking the referee for an explanation. He's the captain. That's his role. He's allowed to do that. I'm sure he said nothing offensive. That was a bit too strict. And we will pay for it dearly."
Turkey picked up six points from their three games in Group F to finish second behind Portugal, with the national side in the last-16 stage of a European Championship for the first time since Euro 2008, when they progressed to the semi-finals.
"We hadn't qualified from the group for many years, so we're very happy," Montella told UEFA's official website. "We managed to overcome difficult situations. We want to enjoy this victory. Starting on Thursday we will concentrate on the next match and aim to continue making history.
Turkey picked up six points in Group F
"Some of our players will not play in the Austria match, and we will feel their absence. We need to recover and we have six days ahead of us. In matches like these, we must control our emotions and be more careful in terms of ball control.
"Austria came out on top of a very tough group (which also featured France and the Netherlands). They are very fast, very fluid, a team worthy of great admiration.
"Only three of the starting 11 today had played at a European Championship before. I hope our success in making it out of the group will relieve some pressure."
Turkey were knocked out in the group stage of both Euro 2016 and Euro 2020, with their run to the semi-finals of Euro 2008 proving to be their best-ever performance in the competition.
Should Montella's side navigate their way past Austria in the round of 16, they would take on either Romania or Netherlands in the quarter-finals of the competition.