A quick glance shows two unactive rooks on his side, I give you the edge because the white rooks are menacing.
My opponent said he was winning

That is what I thought. Even without dropping his queen I think that I was winning there positionally

I would give the advantage to you, more active piece play, the kings are pretty much equally safe (his king isn't really safe either) and he's got no real threats going, only an exposed queen. And his seventh rank is really threatened.

Looks like white is totally winning to me. Positionally the game looks over, and also his king is feeling a little shaky. BTW, after 29.g3, white's threatening the very nice 30.Rxf7! which ends the game (Actually I'm surprised you didn't mention it since you are such a tactical wizard).
I hadn't really analysed the game...other than a quick glance. Explain what you mean by Rxf7.
I'm more of a tactical wizard in training. :) Sorta like harry potter

He drops his queen and then plays until mate. This is coming from the same brain that thinks he was winning. We cannot trust the judgment of this brain.

Only on 18th move, 18.Bf4?! could have been replied by 18...g5! => 19...g4
Everywhere else you dominated the game.

well, if 29...Qb5+ 30.Qd3
30...Qxd3 31.Rxd3 looks completely winning for White.
30...Qxb2 31.Rd8+ mates.
and it looks like 30...Qa5 31.Qd6 wins.

I have a suggestion regarding the mating net. Normally, you want to check the opponents king, so that your queen is positioned in the g-file and the opponent's king is on the h-file, after which you bring in the helper rook to check on the h-file. This solidifies the mating net, so that the king can't escape through a safe square on the f-file.
In this game, since you have a rook on the seventh rank, it wasn't necessary, but you could probably have reduced the number of cases you had to think about in ensuring the mate.
So during this game my opponent dropped his queen on move 27 or 28. I ended up mating him, but he said after the game that "it was to bad he dropped his queen because in his opinion positionally he believed he was winning."
I totally disagreed with him saying that I thought I was winning. Here is the game. Do you think black was winning?