Michail Antonio netted decisively after half an hour and West Ham also struck the crossbar in each half through Mark Noble and substitute Andy Carroll.
The defeat leaves ex-West Ham boss Allardyce and his squad in the relegation zone, and despite his solid record during his time in east London, current coach Bilic saw no reason to go overboard with the goodwill noting that he felt his team deserved to edge victory.
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"I don't feel sorry for him – I respect him and I said enough about him at the pre-match press conference, that he did brilliantly for West Ham," Bilic said.
"I don't feel sorry, it's not the right thing. We deserved to win.
"We created more chances. At the end of the day we scored the goal, we hit the bar twice.
"We were, more than a few times, in situations where we tried to be too cute and we didn't want to be lethal.
"Until you score the second one… this is the Premier League. There is always the danger that you are going to concede.
"Yes, they created a couple of chances and they played some good football in the second half but if you are trying to tell me that they deserved to win this game more than us, no way."
Bilic reserved praise for matchwinner Antonio, who scored for a third consecutive home match having had to bide his time and wait for a first-team opportunity since joining from Nottingham Forest in September
"He's a good example for this new generation of players who are expecting first to get a chance and then they are going to show you how good they are. They don't need to deserve a chance," he added.
"He probably dreamed about the Premier League for a few years at Forest or before in non-league football.
"Then he got it in the end. He waited for his chance because at that time we had some players in front of him. He continued to train hard and when he got a chance he used it, big time."