You do not need to have the stuff on hand so it is not the p that saves your opponent. If your opponent gets a n, b, r or q he/she will avoid mate and they therefore do not consider it as mate. If your partner is up on time and sits you will win the match.
I played Qh1 and then my partner was mated a few moves later on their board. Can you put a pawn on the 1st rank as white? What am I missing?