A goal was still all Celtic needed to go through but in the 61st minute striker Aleksandr Kokorin made it 3-0 on the night to end the Scottish champions' hopes of reaching the last 16.
The Scottish champions found themselves behind in the tie before the half-hour mark, as Daler Kuzyayev crashed a swerving, long-range strike past Celtic goalkeeper Dorus de Vries.
"But I just thought defensively, as a back line, we didn't press the game or squeeze the space".
Branislav Ivanović had a goal and an assist to help Zenit Saint Petersburg beat Celtic, while RB Leipzig beat Napoli on aggregate despite losing their second-leg match.
Speaking to reporters after the game, the former Liverpool coach said: "In this one, if we had played with a bit more courage and belief, we maybe could have got a result".
"As always, we like to play with aggression and with that educated pressure on the ball - if you can press, great, if not, make sure you're super compact and the look to go from there and create opportunities".
The Russians may have drawn a blank in the first leg, but they scored 17 goals in the group stage - more than any other side - and they will feel confident of turning the tie around.
"There's still an very bad lot of work for us to do, but taking a step back, there have been a lot of positives for us in both European competitions". We are taking steps forward with a lot of young players.
"We deserved to win", said Mancini.
"We know that over the course of the two legs we need to replicate that but the players are ready for that". At the beginning of the first leg, I was being a realist as everyone would probably expect Zenit to go through, but what we showed in the first leg is that we're going to make it very hard for them and going to make it really competitive.
Despite the defeat to Zenit, Celtic still have plenty to play for this season, and look on course for another treble.