The English side had hit the woodwork twice and seen several shots whizz just wide before falling behind to an Edinson Cavani goal midway through the second half as Napoli lived up to their reputation as ruthless counter-attackers.
City coach Roberto Mancini blamed first-night nerves for the mistakes, which included having to call on captain Vincent Kompany to clear a Marek Hamsik shot off the line, and he was relieved to get Kolarov's 75th-minute equaliser.
"All the players wanted to win this game and to score. Instead if we played simple we would have done better," the Italian told a news conference after his side failed to match their breathtaking start to the Premier League season.
"It was maybe desire or the pressure... taking forward seven or eight players is just not going to work," added Mancini, whose City side are joint top of the Premier League with maximum points from their opening four games.
With Bayern Munich and Villarreal the other opponents in Group A, Napoli had been widely seen as the most straightforward opponents for the world's richest club even if they did finish third in Serie A last season.
City had hidden their nerves well with a lively start as Samir Nasri tried his luck in the third minute before Edin Dzeko used all his muscle and height to power through the Napoli defence only to send his effort just wide.
The hosts got a sharp reminder to be wary of their visitors on the break midway through the first half when forward Ezequiel Lavezzi cracked a powerful curling shot against the crossbar.
But they did not take note and after comfortably controlling the first half, City gave their visitors more of the ball and more space in the second period.
Napoli midfielder Christian Maggio was allowed to charge a long way up the pitch after City lost the ball in midfield and he fed Cavani who kept a cool head to slot through keeper Joe Hart's legs to give his side a 69th-minute lead.
"I told the guys to pay attention because Napoli are very good on the counter attack but we lost a stupid ball in the middle," Mancini said.
Shortly before that City had been lucky when another lapse allowed the lively Hamsik space on the right side of the goal and his volley beat Hart before being hoofed away by a quick-thinking Kompany.
With Mancini having already criticised his side for squandering numerous clear-cut chances in Saturday's 3-0 win over Wigan Athletic in the Premier League, he witnessed many more examples on the European stage.
Midfielder Yaya Toure clipped the underside of the crossbar with a drive after a move he had started in his own half before the break while forward Sergio Aguero, who scored a hat-trick at the weekend, smashed a shot at the woodwork later on.
Just when it appeared City's first venture into the European Cup since 1968/69 was turning sour, Kolarov stepped up for the rescue mission despite earlier breaking his nose.
He launched a free-kick straight over the wall from 25 metres out and sent the ball flying into the top corner to send fans into jubilant celebrations.
Napoli boss Walter Mazzarri was delighted with the way his team had played but admitted that, like City, they had also had the jitters because of the occasion.
"Obviously in the first few minutes we were a bit tense," he said. "In the second half we were in much better shape. As soon as we were one goal up I was very happy. [But] we could have gone one further."
The match was being watched by Mancini's father, who the Italian said would have plenty of views to air.
"I await his criticism tonight," he said.
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