Trailing 2-0 to 10-men Everton - who had Kevin Mirallas sent off in the first half - after goals from Romelu Lukaku and Aaron Lennon 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 - and Bilic - into rapture.
"I didn't know what to do
I went down [the tunnel] to calm down and came back straight away," he said.
"It was a great comeback, a great game
To do it like this is unbelievable.
"Of course they are going to cause you problems with Lukaku, Lennon, (Ross) Barkley but when we had the ball I always had the feeling we could do it.
"I said at half-time 'We are one down but we are going to do it
We have to be less sexy and more lethal in front of the box'.
"We knew it was not going to be one-sided match when you have a player like Lukaku.
"When they had the ball, he makes that one-player loss less visible
He creates mayhem.
"To come back like this nobody can deny we didn't deserve it
We showed our quality, we showed our stubbornness and we got a great win."
The turning point was when goalkeeper Adrian saved Lukaku's 69th-minute penalty which would have put the result beyond West Ham.
But from that moment they were resurgent and Everton failed to deal with everything their opponents threw at them.
"If they had scored that penalty they would have won the game maybe 3-0 or 4-0," added Bilic.
"What I am pleased the most about - apart from three points - is we never lost our composure, our patience."
Everton manager Roberto Martinez refused to blame either Mirallas, who saw red for a second bookable offence after earning an initial yellow card for diving, or Lukaku for his woeful penalty attempt.
"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," he 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."
Source : PA
Source: PA