A controversial Mark Noble penalty and a James Chester own-goal ended the London club's run of three consecutive defeats and ensured they took a giant stride towards preserving their Premier League status.
The victory moved West Ham up to 11th in the top flight, nine points ahead of third-bottom Sunderland and just four points shy of the 38-point target set by Allardyce to stay up.
|WEST HAM LATEST
|20/1||West Ham are 20/1 with Paddy Power to be relegated|
Allardyce's side played the majority of the game against 10 men after goalkeeper Allan McGregor was shown a straight red card for upending Mohamed Diame.
That decision led to Hull protests, as referee Mike Dean only pointed to the spot after being consulted by his assistant and Diame appeared to handle the ball before being clattered by the Scotland goalkeeper.
Noble made no mistake from the spot, but the home side were booed off at the break and Hull equalised early in the second half when Tom Huddlestone's free-kick went in off Nikica Jelavic.
West Ham had another stroke of luck when Chester scored an own-goal and despite hanging on for all three points, the Upton Park faithful voiced their disapproval at the full-time whistle, much to the manager's frustration.
Allardyce cupped his hand around his ear in the crowd's direction before heading down the tunnel, and vocalised his disbelief afterwards.
"I've never been a place where I won and got booed." the former Blackburn Rovers manager told BBC Radio 5 Live.
"I started at 16, got into the first team at 18 and I'm 59 now and I have never been in place where we have won and got booed.
"At half-time players were talking more about fans booing them than the game. Fans affect players. We don't need them on the players' backs when we are coming off three defeats. They have to stay and help them win.
"They are frustrated. I get frustrated. I want the players to play better and take advantage more of the 10 men, but you don't at this level.
"People are organised, wait for an opportunity to suck you in and expose you, even with 10 men. It's not an easy task and we should all be very grateful."