Barca have been eliminated in the quarter-finals of the Champions League in the past three seasons, but appear on track for a better run this campaign.
Messi, who has netted five times in the competition this season, is hungry for a fifth Champions League crown – but he wants even more than that.
"The Champions League is the dessert for everyone, but to get it you have to be good in the league and cup," he told Movistar.
"You have to be good in the league to be well in the Champions League. We will not leave aside any competition.
"The Champions League is special but we will fight for the league and cup."
Barca ended a three-match winless run with their victory, in which Philippe Coutinho and Ivan Rakitic also scored before Messi's double.
Messi hailed his team's display, particularly in the first half, saying: "We were aware that with the results we had brought, there were doubts.
"It's never easy to play as a visitor in the Champions League and less against a team like Tottenham.
"We did a great match. The first half was extraordinary and we are happy because everything went as we wanted."