The Reds failed to test Zenit goalkeeper Vyacheslav Malafeev in the first half, with Luis Suárez being presented with three clear cut chances to put the visitors ahead.
Liverpool seemed to be heading for a stalemate before former Porto striker Hulk fired home a scintillating 30-yard strike to put the Russian outfit ahead with 20 minutes remaining.
Matters then worsened for the Merseyside outfit after slack defending by Glen Johnson allowed Sergei Semak to finish from close range, meaning Liverpool must overturn a two-goal deficit at Anfield next week.
“The next goal in the tie is vitally important,” Rodgers said.
“What gives us great hope is that we created enough tonight to win the game. The atmosphere at Anfield will be fantastic – as it always is – and the fans will get behind the players.
“If we can get [the next goal], it can create momentum for us. There’s no way the tie is over.
“It’s obviously disappointing to have lost the game but we’re more than capable of getting goals back and that is something we will fight to do next week.”
Rodgers also promised that Liverpool would cut out the mistakes in months to come after Jordan Henderson and Johnson were at fault for Zenit’s goals.
“In six to 12 months we won’t be making those mistakes,” added Rodgers. “This is a group that is growing and has come together. And this is part of our growing pains as a group.
“We’ve put in some terrific performances this season and not always got the results. We have to be more clinical and that is something that, over the coming months, will certainly take place.
“For the first five or 10 minutes their winger Danny was giving us a problem in behind the midfield. Once we organised that, I thought we looked solid and we were a threat going forward.
“Arguably, we could have been leading at half-time. It was important for us to start the second half well and that is what we did. Going into the last 20/25 minutes, I thought we could go on and win the game.”
MORE: Latest Man Utd news
MORE: Latest Arsenal news
MORE: The latest football news