Solution:

As @iceblue found out (congrats!), the best attacking move was

1. Rh3.

Now, after 1. ... f5 white would reach a very good position with 2. Qh5 Nf6 3. Bxf6 Rxf6 4. Qxh7+, and after 1. ... h6 simply followed 2. Bxh6!.

In the game my opponent replied with

1. ... g6.

However, I could win with

2. Rxh7!

The game continued with

2. ... Kxh7
3. Qh5+! Kg8 (3. ...Kg7 loses even faster because of 4. Qh6+ Kg8 5. Rf1!.)
4. Bxg6!

4. ... fxg6
5. Qxg6+ Kh8
6. Rf1! with the idea Rf3 -> Rh3# checkmate.

Now there are two possibilities:

a)

6. ... Bxf4
7. Bxf4 Qd8 (In the game black played 7. ... Rf6 but resigned after 8. Be5 Rf8 9. Rf3!.)
8. Qh5+ Kg8
9. Rf3 Rf7
10. Rg3+ Rg7
11. Rxg7+ Kxg7
12. Bh6+ Kh7
13. Bg5+ Kg8
14. Bxd8 Rxd8
15. Qg5+ Kf7
16. Qxd8 and white wins.

b)

6. ... Ne5
7. Bf6+ Rxf6
8. Qxf6+ Kh7
9. fxe5 Rf8
10. Qh4+ Kg8
11. Qg3+ Kh7
12. exd6 Rxf1+
13. Kxf1 and white wins.