Blues owner Abramovich's quest for beautiful - as well as winning - football has led him to dispense with a string of managers, with Benitez the latest to be charged with delivering the Russian's dream.
But the sacking of the Champions League-winning Roberto Di Matteo and appointment of former Liverpool boss Benitez as interim manager has infuriated Chelsea supporters.
Visiting fans were heard to chant, "Roman Abramovich, is this what you want?", during Chelsea's 3-1 Premier League defeat at Upton Park.
But Hammers boss Allardyce, himself no fan of Benitez, said: "Roman is the owner of Chelsea Football Club and he's entitled to do whatever he wants because of the amount of money he puts behind the team.
"He's been a great advocate for why the Premier League has become such a great brand around the world of football, by the money he's put in. What he wants, you have to deal with if you're the manager. And you have to deliver."
Allardyce admitted he exploited Chelsea's weaknesses on Saturday, especially in the absence of their injured captain, saying: "I think they're missing John Terry. I think they're missing leadership."
But he added: "Look at the amount of money they've spent on the players and you can see the quality they've got.
"But, in the second half, they didn't quite show as much resilience as they needed to, to overcome what we were doing.
"They couldn't change the way they played to cope with us.
"We changed the way we played to cope with them and expose them. They couldn't do that against us in the second half."
West Ham's onslaught was spearheaded by former Chelsea striker Carlton Cole, who filled the void left by a serious knee injury to Andy Carroll.
Cole has been in and out of the side this season but now looks set for a lengthy run up front, with Carroll set to be out for up to eight weeks.
"I'm always there and always have been there, for the back-up and always ready to do my best when called upon," said the 29-year old, who scored West Ham's equaliser yesterday, his first goal of the season.
"Andy has done a great job for us, he has done brilliantly. He scored against Tottenham last week and wanted to keep that going, but he has picked up an injury.
"I have perhaps not had a good time in the Premier League this season until now, but I knew if I kept working hard and training hard, the goal would come."