Arsenal were reduced to 10 men when Per Mertesacker brought the striker down when he was through on goal in the 18th minute, before Costa gave the visitors the lead shortly after.
The Gunners struggled to gain a foothold in the game and eventually slumped to their second loss of the season at the hands of the west London side.
And Arsenal chief Hill-Wood was furious at Costa after the final whistle as Chelsea dealt a significant blow to the north London side’s hopes of winning their first title since 2004.
Hill-Wood is quoted as saying by the Daily Star: “I think he deliberately ran in front of Mertesacker, who hardly touched him.
“Yet Costa rolls around as if his legs have been cut off. Once the red card’s been shown, he’s suddenly perfectly all right again.
“That’s because there’s absolutely nothing wrong with him. There certainly wasn’t soon after when he scored that goal.
“Personally I think the referee should have sent him off for diving. It’s very annoying because to me it is blatant.”
Arsenal’s next Premier League game is a home clash against Southampton at Emirates Stadium on 2 February.
Before that, the Gunners host Burnley in the fourth round of the FA Cup this weekend.
MORE: The latest football news