A first-half goal from Danny meant the Russian champions claimed the win at the expense of the Rossoneri, and third place in Group C, but Ambrosini was adamant his side had done enough to emerge victorious at San Siro.
"We talked about what we should have done on the pitch before the game, we tried to play the best we can as usual," he told reporters.
"We're through, anyway, but I think that the round of 16 will be really tough. We are improving match by match, and we can do even better than what we have accomplished lately.
"We lost tonight, but I think that we played better than them. We played a good first half, maybe we didn't create many chances but we were in control of the ball and of the game.
"We're sorry that we lost, we wanted to keep our momentum going, but that's football."
Meanwhile, Ambrosini's team-mate Mathieu Flamini shared his sentiments, claiming Milan had "dominated" their opponents but failed to reap any rewards for their efforts.
"We took the initiative and created many scoring opportunities. They defended well and didn't allow us anything," the Frenchman added.
"The president [Silvio Berlusconi] told us this was an important game and we wanted to win, but were unable to do so.
"We are disappointed because we dominated the game, but instead they punished us with their only chance in the first half.
"After the goal they shut up shop and we couldn’t find the space to score despite our efforts."
Milan's next match is against Torino on Sunday, as they aim to make it three wins on the bounce in Serie A.