But little did Elliott know, his shot had taken a deflection to stop it from going wide, and was ultimately put down as an own goal by Hugo Bueno instead.
Liverpool went behind but turned a dismal first half around to win 3-1, with Elliott's effort finishing the job in added time.
The England international got on the end of a brilliant low cross from Mohamed Salah as the game ticked over 90 minutes, sending the away end wild.
"Elliott hits it like an arrow," ex-Arsenal star and talkSPORT's commentator for the game Martin Keown said. "A really good strike."
Elliott picked up a yellow card after jumping the barriers around the pitch to embrace the celebrating Liverpool fans in the away end.
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The midfielder went crazy with the jubilant travelling supporters, soaking in what he thought was his first goal of the season.
However, despite the 20-year-old thinking he'd now scored at Molineux in consecutive seasons, it actually went down as an own goal following the deflection from Wolves' Bueno.
Goals from Cody Gakpo and Andy Robertson to cancel out Hwang Hee-chan's opener mean Liverpool still walk away with all three points, though.
The win takes the Reds to the top of the league, leap frogging Man City who are in action against West Ham this afternoon.