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The fourth-round replay was seconds away from a penalty shootout when Angelo Ogbonna's header made it 2-1.
"It will go into the history of West Ham's greatest games," said Bilic. "Against Liverpool, the last season at Upton Park, night kick-off, a goal in the last minute.
"They left their heart on the pitch."
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The Hammers had never beaten the Reds in the FA Cup and have not won the trophy for 36 years. They will face Championship side Blackburn in the fifth round on Sunday, 21 February.
Bilic, 47, said earlier this year he would rather win the cup than finish in the top four of the Premier League and qualify for the Champions League, saying "the proof is silverware".
After Italy defender Ogbonna's added-time winner, the Croatian boss said: "I'm a down-to-earth guy, but tonight was a big, big match and I was a bit emotional at the end of the game, I'll admit it.
"The players made the fans proud of them and we want to progress even further. If we produce a performance like this then we are capable of beating any team in England."
Bilic also said midfielder Dimitri Payet is committed to the club after reports linked the France international with a big-money move to China.
Payet, 28, is in talks over a new contract and Bilic said: "We are a club moving forward and the most important moves are on the pitch.
"We must do everything we can to keep our best players - and he is our best player. He's happy here and very soon we will sort out his situation."
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Liverpool manager Jurgen Klopp felt his much-changed side should have won the game after Philippe Coutinho's low free-kick had cancelled out Michail Antonio's opener.
The Reds' £32.5m striker Christian Benteke again missed several good chances as the visitors had 20 attempts on goal - one more than West Ham.
"It's hard. We were the better team, created chances and played good football, but we didn't use them," said Klopp.
"There were lots of decisions that on a better day I'd say were funny decisions.
"It's not easy because we were only here to win. You don't have to show your best, you just have to go to the next round.
"The door was wide open for us because we did well, but it was not enough. We have to carry on and fight to the next game."
West Ham's extra-time victory over Liverpool was "a classic FA Cup tie" and one of the club's best, according to manager Slaven Bilic.