An early goal for the Russians through Hulk left the home side needing four goals to progress to the last 16 stage of the Europa League.
Backed by a raucous atmosphere to rival some of the club's most famous, Luis Suarez converted two free-kicks either side of a Joe Allen finish - although the Reds were ultimately eliminated on away goals.
After the encounter, the Spaniard revealed his pride at the support he and his teammates received.
"Thanks again - they've been brilliant, as they always are," Reina told Liverpoolfc.com.
"They were with us even more when we conceded the goal. That's something to be very happy about and I feel very proud.
"We're disappointed because we didn't get through but we have to be proud of what we did.
"We went out with our heads held high. All we could do was try our best and that's exactly what we did.
"We showed great character, especially after the goal - we just gave our best more than ever. We fought during the 90 minutes for our lives. We showed passion out there.
"We're not happy because we went out but that's football."
Elimination from continental competition means the Barclays Premier League is now the only remaining front on which Liverpool are fighting.
Reina insisted that victories will continue to be the primary focus before the campaign concludes and hopes the squad can benefit from added freshness during the final stretch.
He added: "To win and win is the only target. We need to go step by step, try to climb up the table and make sure we win as many games as possible.
"We can only focus on the next game, which is Wigan.
"Maybe when you are in both competitions at the same time you somehow play better. Hopefully it will help the team and we will at least be fresher."