Solution:
Congrats to @theaustrianguy for finding the key move (and thus being the winner of this round)!
The first move is
1. Ra4! (idea: 2. Rxa7 Rxa7 3. Rd8# checkmate) after which white has a clearly better position in every variation.
My opponent responded with 1. ... a6 (moves like 1. ... Ne7 or 1. ... Kf8 had been somewhat tougher but not sufficient, as well) - see position below:
I guess he didn't expect the answer
2. Rxa6!.
He continued with 2. ... Rxa6 which allowed me to checkmate him immediately:
3. Rd8# checkmate (the knight at c6 is pinned):
However, also 2. ... bxa6 wouldn't have helped because of
3. Bxc6+ Kf8
4. Bxa8 and again black cannot prevent the checkmate:
Actually, the rook swing from one side to the board to the other was rather creative and aesthetical.