Despite dominating throughout, Juventus fell behind when Mikkel Beckmann lashed an excellent free-kick past a helpless Gianluigi Buffon in the 50th minute.
The Italian side continued to pile on the pressure but a combination of excellent last-gasp defending and poor finishing looked like it may secure a shock win for Nordsjaelland until Vucinic lashed home from close range nine minutes from time.
And while Alessio felt his side were more than worthy of a share of the spoils, he admitted their profligacy nearly came back to haunt them.
"We created so many chances, at least 14 or 15, but we couldn't put the ball in the back of the net. Thankfully Mirko got the equaliser at the end, otherwise we might well have suffered a shock defeat," he told the club's official website.
"We lacked a cutting edge in the first half, especially in front of goal. We were much better in the second half and the lads performed well.
"It's a shame we've only drawn but we need to roll up our sleeves, keep going and try to win our coming games."
Juventus have won seven and drawn one of their opening Serie A matches but they have been unable to maintain that momentum in the Champions League.
Last night was the third time in as many matches they had to come from behind to claim a draw in Europe's premier competition and Buffon has demanded improvement.
"I'm angry both about the result and the way their goal came about," said the Juventus captain.
"We gave away a needless free-kick and they were only going to score from a set-piece or a goal-mouth scramble. The shot was perfect and there was nothing I could do.
"The first half was even but there was only one team in it in the second period and the result is not a true reflection of the game.
"What's different compared to the league? This is a step up, what might be sufficient in Italy is not enough in Europe. There are no easy wins in the Champions League.
"Nordsjaelland matched Chelsea for the best part of 75 minutes. Now we must do everything in our power to win our remaining fixtures."