Mirallas saw red for a second bookable offence after earning an initial yellow card for diving and Lukaku had a woeful penalty saved in the second half.
Trailing 2-0 to ten men, the Hammers responded with three goals in the final 12 minutes.
Michail Antonio and substitute Diafra Sakho struck within four minutes of each other before Payet popped up in the six-yard area to send the travelling support into raptures.
“We were presented with a very difficult game when Kevin had the red card but you look at the way we adapted, worked for each other, and we were the better team,” the Toffees boss said.
“You wouldn’t know West Ham had an extra man.
“He (Mirallas) apologised but it is part of the game. He didn’t do anything apart from trying to play the game.
“The penalty was a big blow psychologically. Romelu is such a reliable footballer that we were all a little bit shocked. Those were two key moments.”
Martinez was more inclined to blame referee Anthony Taylor, although having already booked Mirallas he was given little choice when the Belgian needlessly clattered into Aaron Cresswell.
“The second booking is a decision the referee has to make. The first one is a ridiculous decision,” he added.
“Why should we book a player because we feel it is not a free-kick? Two players come together, they cross each other and he loses his balance.
“I don’t feel he is a referee who understands the game in a way that we want the game played.
“You want a referee who takes the backseat. Unfortunately we had him here against Swansea and he had a very unfortunate game.
“You want to see refs who understand the game and allow players to play and not just have to apply the letter of the law.”