Opposite color bishops

My oponent placed too much confidence in drawing opposite color bishops.

Next time it might help if you bothered mentioning a few more minor details, like what color you were playing - instead of letting everyone else having to figure it out on their own. You could have just cut and paste the pgn file into the board editor here, which would have inc all the game details (I usually delete my opponents nm if I won)

He (she) shouldn't have resigned so soon, (as you noted) your 2 connected passed pawns do look imposing but opposite-color Bishops can be very annoying sometimes even vs such imposing odds.

You should submit the game for analysis in addition to doing some of your own. 

Your game is actually similar to a game I just went over, Korchnoi-Botvinnik 1960.

To begin, here's a basic drawing method followed by an example of when it doesn't work. 

After that is another way you can win with opposite bishops.  These are useful to know when looking at the game example of how Korchnoi wins and also in the variation of how he could have won if Botvinnik had improved his defense.

In the above example if we switch the color of the bishops then black cannot prevent the winning move g6.  This is the position Botvinnik found himself in during the game.  In your game white has the same opportunity / winning plan.

But first, the other common way to win is if the pawns are far apart.

Here's how Korchnoi did it (create passed f and g pawns).  Notice move 36 threatens to create a passed b pawn so the bishop must abandon its blockade.  Botvinnik suggested the variation after the game.  Later Korchnoi found a win from the variation.  Comments are Korchnoi's.

http://www.chess.com/echess/game?id=63734484

This is good review for me.  I actually never noticed that in certain positions if you switch the color of the opposite bishops the game can go from drawn to completely lost (no matter the formation!)

As long as you keep a pair of major pieces on the board, the opposite color bishop disadvantage is less signifcant. You don't want to trade into a solely endgame of bishops of opposite color unless you have at least have a 2 pawn advantage+ positional advantage, or just an outright dominating positional advantage.