"We need to really, really fight. It will be a lot harder than last year, a lot, lot harder," he said. "A hard path became so much harder because of two defensive mistakes. We shouldn't despair, we have four games left, we have to grit our teeth. If we're good we'll get through."
Next up for Juventus is a double header against Real Madrid, which beat Galatasaray 6-1 in its opening match and won 4-0 at home to FC Cophenhagen on Wednesday to take firm control of Group B.
"I don't think Real Madrid will beat Juventus twice," Mancini said. "I've seen them twice recently and don't think they're in form really. We'll try to do very well against Copenhagen but we need to improve on tonight and play better. I think the group is still very open."
Former Inter Milan and Manchester City coach Mancini was confirmed as Fatih Terim's replacement at Galatasaray on Monday and only met the squad shortly before their flight to Italy.
"The only advantage I had was that of knowing Juventus," Mancini said. "I didn't know my players so had to get to know them quickly, had to learn their names quickly. I would definitely have signed for a draw before the match, even though I always think you should go for a win.
"We were unlucky with the penalty, I don't think it was a penalty. We defended well and it's not easy to defend against a team like Juventus, then the game changed with the penalty."
Both teams had endured a disappointing start to their European campaigns and they started cautiously in Turin. Juventus had the first real opportunity in the 15th minute, but Paul Pogba slipped as he attempted an audacious overhead kick and missed the ball completely.
That seemed to spark the game into life and Drogba almost put Galatasaray in front four minutes later with a shot from a tight angle which Juventus goalkeeper Gianluigi Buffon cleared with his foot.
A setback followed shortly afterward for Juventus when Mirko Vucinic beat the offside trap to race onto a delightful chipped pass by Andrea Pirlo, but stumbled and injured himself in the process.
Vucinic was replaced minutes later by Quagliarella, while Galatasaray defender Semih Kaya made way for Gokhan Zan.
Quagliarella had an opportunity with his first touch in the 28th minute as he prodded a cross narrowly past the left post. The Juventus forward was caught up in a tangle with Zan as he struck the ball but the referee rejected his furious claims for a penalty.
Galatasaray broke the deadlock when Bonucci woefully misplaced a back pass, giving it straight to Drogba, who easily bypassed the outrushing Buffon to slot into an empty net.
Juventus could have leveled quickly but Carlos Tevez's fierce shot scraped the outside of the right post moments later.
And Galatasaray could have extended its lead when Wesley Sneijder was left completely unmarked and blazed well over the crossbar from just inside the area.
Arturo Vidal went close as he twice fired narrowly over the crossbar in quick succession.
Conte brought on Fernando Llorente for his Champions League debut and the Spaniard almost headed Juventus in front moments later but his effort also went just over the crossbar from close range.
Juventus did level in the 77th minute when Quagliarella made the most of a mistimed challenge from fellow substitute Amrabat and Vidal blasted home the spot kick.
Llorente had the ball in the back of the net in the 84th but it was ruled out as Tevez was offside in the buildup.
Juventus had come from behind to win two of its previous three league games and it looked like doing so again when Quagliarella headed home Pirlo's cross from close range four minutes from time.
However, a Galatasaray counter attack led to an 88th-minute equalizer as Drogba took advantage of gaps in the Juventus defense to pick out an unmarked Bulut on the far side of goal.
Llorente missed a chance for Juventus in stoppage time when he toe-poked the ball wide of the right post.
