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Since the a2 rook confines the black king to row 1, I figured I had to put the queen on c1. The key is to move the bishop off the c column to allow the queen to move to row 1. That requires the bishop to move to a7, b6, d4, or e3 so it can keep an eye on the b1 square.

The catch is the black rook can move along row 7 to block the white queen once the queen goes to row 1; and that would result in mate in three instead of just two moves. This meant the bishop could not go to either a7 or b6 since neither of those can block the black rook if it moves to d7, e7, or f7. That left either d4 or e3 for the bishop move.

Should the black rook move along row 7, the queen can also go to h3 to deliver checkmate; but not if the rook blocks the queen's diagonal move by going to d7. But if the rook is on d7 to prevent the queen from going to h3, then the queen can still checkmate in 2 by going to c1. And that requires the bishop to be along the d column to stop the rook. Thus, the only square the bishop can go to is d4.

I saw that the first move must be a bishop move in just a few seconds, and then spent a few more seconds using a process of elimination to settle on bishop to d4.

... and your logic is OK. I think my logic was quite OK too, but it took me about 3 minutes. I ended up on the same address, but i walked a few blocks more  

After trying some options, i reasoned that the queen should come in a position where it can reach the 1st row or the h-line (depending on how the black rook blocks it). Since the queen may not leave the 8th row (because of Rh8+), i tried Qd8, Qe8, Qf8 and Qg8. But in all these cases the rook can block the queen in such a way that it requires 3 moves for a mate. For instance after Qe8 black plays Rf7. Since h5 is blocked now, white has to play Qe1+ and then the rook blocks it again ...

Ultimately i reached the conclusion that the queen is fine where she is and that we have to move the bishop. So i solved it the hard way  

what...
pls do use the ALTERNATE SOLUTION option if you cant see it to check whether you really found a solution to the puzzle. tired of ppl unable to use that option -.-

1. Bb6 is followed by ...Rd7 btw with no mate following in the next move for white.

not good there are other way to mate intwo moves..

@yadasampati put Q to e8 where ever you put your rook after that you are mate...that is mate in two moves how do you explain that?..

Well, others have already informed you, but i would still like to mention that i had already given the example of Qe8 in a comment posted before yours ... (titled "30 seconds is fast ...")

I literally say the following: >> For instance after Qe8 black plays Rf7. Since h5 is blocked now, white has to play Qe1+ and then the rook blocks it again ... <<