In endgames there are always exceptions, even to the exceptions! Let me show you what I mean. The position to the left is with black to move. If you showed this problem to a non-club player or even a club player without end game skills, the patzer would quickly say "Black should win this easily, he's up a whole queen!"
Poor foolish know-nothing patzer. Every chess player with an ounce of end game knowledge knows for certain this is a draw. What the patzer does not realize is even though he is in fact up an entire Queen, a rook-pawn on the 7th and a bishop-pawn on the 7th rank can actually draw versus that queen. Hence this is an exception to a Queen being better than a pawn.
1... Qg2+ (any Queen move that remains on the B file is an instant draw by stalemate) 2. Kb8 Qg8+ 3. Kb7 Qf7+ 4. Kb8 (4... Ka6?? 5. Qa8! -+) Qe8+ 5.Kb7 Qd7+ 6. Kb8 (6. Ka8?? 7.Qc7#) Qd8+ 7.Kb7 =. Black not make any progress because if he tried to move his King up to aid in the fight white answers with an immediate a8=Q and white draws.
Now here is the funny part, in endgames there are always exceptions... even to the exceptions. In this instance it turns out the silly patzer is right! This is a forced win for black! I just showed you how black throws away the win by playing 1. Qg2+ or 1. Qh7 with similar ideas. The trick here is to bring the black King closer with tempo. Thus the subtle 1... Kb3! wins.
1... Kb3! 2.Kb7 Kc4+! 3. Ka8 Qh2 4. Kb8 Kb5! (go ahead, make a Queen, see if I care) 5. a8=Q Kb6! (that's what's up white, now what you gonna do??) Here white has nothing. 6. Qb7+ loses to 6. Qxb7#
The inherent lesson here is not necessarily mechanical, moreover it's a lesson in dogma. There is absolutely no room for rigid thinking in chess. How foolish would you feel if you agreed to a draw in this position only to plug the game into Fritz later to have him tell you, "HAHAHAHA you fool! You agreed to a draw in a won position!" I am not necessarily saying that it would have been easy or even possible to find this continuation over the board, it's certainly a tricky one. If you had shut down your brain when presented with the problem you have no chance at a win at all. When presented with this kind of problem over the board instead of saying, "Oh rook-pawn on the 7th, draw." Force yourself to work out the continuation in your mind as to why it is indeed a draw. Look for some crazy variations that give your opponent a real chance to stumble into a mistake, then if all roads lead to a draw only then should graciously offer.