The midfielder picked a timely moment to score his first Premier League goal of the season to sink Swansea.
Bilic has been feeling the heat after five straight defeats left West Ham in serious danger of being sucked into the battle at the bottom.
A sixth loss would have left them just two points ahead of Swansea, and possibly signalled the end for Bilic, but instead they can breathe easier after moving eight points clear of the drop zone.
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West Ham: Randolph, Byram, Fonte, Collins, Masuaku, Kouyate, Noble, Antonio, Lanzini, Snodgrass, Ayew
Subs: Nordtveit, Feghouli, Carroll, Adrian, Sakho, Calleri, Fernandes
Swansea: Fabianski, Naughton, Mawson, Fernandez, Olsson, Fer, Cork, Carroll, Sigurdsson, Ayew, Routledge
Subs: Amat, Ki, Nordfeldt, Montero, Narsingh, Kingsley, Llorente
Referee: Kevin Friend (Leicestershire)
WHAT THEY SAID
Slaven Bilic: "I've seen bigger clubs in trouble and then when they stay up they celebrate like hell.
"The expectations were to make a big step up, but suddenly we are in a situation where we need points.
"The expectations were that it was going to go down in history as a great season, but it ain't. It's either going to go down as a negative season or just okay. But we're not going down."
Paul Clement: "We're in the bottom three and there are seven games to go, and West Ham are a team who have had their own troubles recently, so we have to go there believing we can get a result.
"We need to get a result, especially in the next two games because the games are running out now."