Carroll was dismissed in the 27th minute after collecting two yellow cards in the space of 99 seconds – both times for using his elbow.
The Hammers were ahead at the time through Michail Antonio’s opener but the 10 men were unable to hold out and Chris Wood earned the Clarets a late point.
Bilic said: “I have no complaints with Andy’s sending-off. I’m very disappointed and angry.
“With the second one you can’t do that, especially two minutes after you’ve been booked.
“It was an unbelievable bad decision. Losing Andy was a big blow for us.”
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West Ham’s next match is with Brighton at the London Stadium next week.