The one-goal lead from the first leg of the last-32 clash was wiped out before half an hour had been played as defender Branislav Ivanovic put Zenit in front with a header.
Daler Kuzyayev fired in a long-range drive in the 27th minute to turn the tie in the hosts’ favour.
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.
Manager Rodgers said: “Defensively we lacked aggression, particularly in the first half. We didn’t close the space quick enough.
“We talk about defending forward and we just didn’t defend forward well enough. We conceded poor goals from that.
“The corner is one that can happen, we were blocked off, they worked it well and found the space. Branislav is outstanding in the air.
“The second goal, we didn’t engage quick enough from the throw-in. We’re too deep and have to press the guy who was shooting.
“At 2-0 we still have an opportunity in the game. But everything was too sideways and backwards and that’s about bravery and having that courage to play.
“With so many young players in the team, you see the contrast from last week.
“We started the second half OK but, again, we concede possession too cheaply and they work the ball round for a cross.
“We should never concede that type of goal, the third goal.
“We were much better than them in the first game. If we played with a little bit more courage and belief tonight, we could maybe have got a result.”
Zenit boss Roberto Mancini claimed his side earned their place in Friday’s draw.
The Italian said: “We deserved to win. We played a very good game. Celtic are a strong team but we played very, very well.
“Celtic’s tempo in Glasgow was very difficult for us. Tonight we scored three goals, had other chances and for 75 minutes didn’t concede any chances to Celtic.”
Lunev, Ivanovic (Smolnikov 87,Mammana, Criscito, Mevlja, Kranevitter, Rigoni (Driussi 84),Paredes, Kuzyaev, Kokorin, Zabolotny (Erokhin 75).
De Vries, Tierney, Ajer, Simunovic, Lustig, Ntcham, Brown, Eboue (Rogic 46), McGregor (Sinclair 62), Forrest (Musonda 71), Dembele.
Antonio Miguel Mateu Lahoz (Spain).