The Red Devils were absent from the tournament last season after finishing fifth in the 2015-16 campaign.
Jose Mourinho's side ended up winning the Europa League, which resulted in an automatic spot in the group stages of the elite competition.
United will kick off their campaign with a home clash against Basel this evening, before facing CSKA Moscow and Benfica in the first stage of the tournament.
While speaking to United Review, Pogba said: "Obviously, I came to Manchester United to play in the Champions League. I knew, last season, when I signed and came back here, that I would be playing in the Europa League. So the objective was to win that to obviously go back into the Champions League. That's what we did and I'm very happy. Now we're just back in it and back to the best cup competition in the world.
"We have a group now and people say it is an easy one for us in the Champions League, but the Champions League is never easy. You have got to play against teams who want to beat Manchester United. We're just back in it, though, and want to prove to the world that we belong in that competition.
"The difference is it's still big in the Europa League and you still play very good teams, but maybe not the biggest clubs. Now, in the Champions League, we could play Bayern Munich, Real Madrid or Barcelona - those teams who all won the trophy many times. I would say that is where we want to play.
"To be honest, I missed [the anthem]. I missed it a lot last season. Just to hear it again kind of pushed me on to give the best. Obviously, when you hear that music, you just have goosebumps coming. You are excited. Everybody is excited – the fans and the players! It's the best thing you can play in."
United are unbeaten so far this season, having won three and drawn one of their Premier League fixtures.