From your initial post one assumes that you asked for push almost simultaneously. It's obvious that if the other one asks for it while you're already disconnecting or finishing push, he should be told to wait, but otherwise, asking just 5 or 10 seconds earlier should not give you any priority. Put yourself in the shoes of this second crew - how much you would be frustrated waiting 4 minutes for your colleauges beside you to finish push and start taxi.
My "everyone pays" part was aimed at your statement that you were faster. All to often I see instances when a smaller/slower traffic is being pushed back and back in the queue ("give way", "give way", "give way") just because of being slower - they are also users of the airspace, and if we're not talking about a procedural airport, releasing a slower departure shouldn't block it for 10 minutes.