The Dutchman scored the only goal of the game as the Reds opened their European campaign with a 1-0 win over Hungarian side Debrecen on Wednesday.
The goal moves Kuyt on a par with legendary midfielder Terry McDermott, who also netted 12 goals for the Anfield outfit in Europe's premier club competition, as Liverpool's third highest European Cup goalscorer.
The 29-year-old is now just two goals shy of Ian Rush's haul of 14 in the competition as the number of games played in Europe nowadays takes its toll on the record books.
Kuyt has three goals in six games in all competitions this season and is happy to have cemented his importance to Rafa Benitez's side.
Special"I didn't know about the record but I do now," Kuyt told the club's official website.
"It's always nice to be important for the club, especially when it comes to Europe.
"Champions League games are always special, so I'm happy I scored again on Wednesday and now I'll try to keep going.
"What I would like to do is become more and more important for the team and try to do my best. I'm happy with my performances so far this season but I need to keep going.
"I'm feeling really confident. I've played the last two games in a slightly different position but for me that doesn't matter. I just want to be important for the team."
TurbulentLiverpool have won three games on the spin after a turbulent start to the season and face a difficult trip to West Ham this weekend in search of a third successive league victory.
Kuyt said: "Everyone knows we had a bad start, losing against Spurs and Villa, so we had to try and get the confidence back, try to change things.
"That's what we've done by winning three games in a row, and now we're looking forward to Saturday.
"Every game is difficult - everyone knows the Premier League is a very hard league. But we have the quality to win these kinds of games on Saturday."