Anderson was then denied by Dubravka but, after Newcastle failed to mount any sort of comeback, the Brazilian made it 3-0 in injury-time when he bundled past Fabian Schar and fired home.
Benitez's side had chances to equalise at a soggy St James's Park, but a second goal from Hernandez ended hopes of a comeback. They have quality players, but in those circumstances you can't give them the kind of space we did.
"He's had a career in which he was wanted by all the big teams and it's because he has something different. It will be a hard month, but I hope it will be a good one".
Asked about the illness which had hampered the frontman in recent months, the manager added: "It was five weeks that he couldn't move, he couldn't work, not just one day".
"After that, he had that virus".
Newcastle did not win a game until their 11th game of the season. "In the last games, he has been playing 45 minutes, but he has recovered now his shape and I hope that he will continue scoring for our team".
Pellegrini said: "I'm very pleased because I think we played a very complete game".
"It's not good to talk too much about it before the game, but afterwards hopefully we can discuss another win".
Perez headed over from close range after meeting Ritchie's 27th-minute corner and then dragged a shot wide, but his side was living dangerously and Arnautovic might have increased the Hammers' lead in the meantime had either Aaron Cresswell or Anderson been able to produce a better final ball.
Benitez, who again fielded loanee Salomon Rondon in attack, told BBC Sport: "They [West Ham] have spent the money on offensive players and they made the difference".
"I can not complain of the effort of the players or the commitment but again today we had chances, but they scored theirs".
"Every top team needs two or three key players to make things happen, and he can be one of those for us".