A Mohamed Salah penalty and Divock Origi strike saw Liverpool overcome Tottenham 2-0 in the decider in Madrid on Saturday.
After helping the Premier League giants win a sixth European Cup/Champions League crown, Mane – who has been linked with Real Madrid – said the squad would push on.
"We have more things to come," he said, via the club's website.
"We will do everything to win [more] trophies with this squad."
The star attacker added: "We are very proud and very happy to win this trophy. The dream has come true.
"I could not have anything better."
Mane produced a good display in the final, but Liverpool needed some fine stops from goalkeeper Alisson to secure their win.
"I can't describe how good Alisson was – I saw him make incredible saves for us," Mane said.
"We are very happy to have him in our squad. He makes us even better. For me he is the best in the world."
Liverpool's sixth crown has them double the next best total for any English side, and they became the first side since 2010 to win a Champions League final while having less of the ball than their opponent.
But the Reds are still searching for a Premier League title having come up just short in the chase this season, finishing a single point behind Manchester City.
Mane will once again be expected to carry a heavy load for the club next season, with he and team-mate Mohamed Salah having both ended up on 22 goals in the league, tied with Arsenal's Pierre-Emerick Aubameyang as the joint-leaders.
The two have formed one of the most formidable strike partnerships in world football over the last two seasons and, having both scored in Champions League finals, are now two of the five players African players to have scored in the European showpiece, following Rabah Madjer, Samuel Eto'o, Didier Drogba.