Trailing to Oliver McBurnie’s second half striker, David Moyes’ men thought they had snatched an equaliser in stoppage-time when Robert Snodgrass fired home from Rice’s pass.
But, with West Ham celebrating wildly, the goal was checked by VAR and eventually ruled out for a handball by Rice.
The ball was headed against Rice’s arm by a Blades player and, while the rule was correctly enforced, the England international was furious with the decision and VAR’s continual influence on the Premier League this season.
“We are fuming. All the lads are livid. I’ve just watched it back. He has knocked the ball into my hand. If you are running with your arms like that with the rules as they are now it is handball but it is not intentional. It is a kick in the teeth,” Rice said.
“I think it is the thoughts of every Premier League player, not just me, pretty much every one doesn’t want to have VAR in the game.
“There have been so many decisions this season that have been absolutely crazy. They were celebrating VAR like it was a goal and football should not be like that.”
The dramatic conclusion left Moyes to come to terms with a first defeat in the third match of his second spell in charge.
It was a painful night all round for Moyes, who saw keeper Lukasz Fabianski suffer a hip injury before his replacement David Martin combined with team-mate Fabian Balbuena to make the error that led to McBurnie’s winner.
West Ham sit just two points above the relegation zone and Moyes, who looked stunned when the VAR decision was made, insisted the goal should have been allowed.
“We’re disappointed. We scored a good goal and it didn’t count. The defender got his head to it and headed against Declan,” Moyes said.
“Whoever checked it I think got it wrong because it was a really good bit of play by Declan, where’d you put your hands, tie them behind your back?
“It doesn’t change my view. I’m a football man, watched and played it all my life and it looked good enough to me.
“I wouldn’t say I’m a great fan of VAR but we are having to come to terms with it. People who have played football know when it’s not right and tonight is one of those situations.”
It was a rare moment of VAR relief for Sheffield United, who have been on the wrong end of several decisions this season.
Blades boss Chris Wilder, whose side are up to fifth place, conceded his team had enjoyed a lucky escape.
“I feel for David, that’s happened to us two or three times this season. We’ve dodged a bullet late on,” he said.
“As soon as they scored, no one was looking at it but I felt for the opposition.
“It’s the same moan and groans as we’ve had in the past. We would’ve been devastated if we’d have drawn that game.”
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