Murray's 29th-minute close-range effort gave Brighton 1-0 win and took the Seagulls into the top half of the table in their debut Premier League season.
Brighton have lost once in six games and their victory was only briefly threatened when substitute Luciano Narsingh struck the crossbar in stoppage time.
Swansea's season is turning into another sorry affair after last term's close shave with relegation.
This was a fifth home defeat in six games and drops them into the relegation zone.
Goals have been in short supply and Tammy Abraham, called into the England squad for the first time this week, was again left too isolated in attack to make an impact.
The growing anxiety around the Liberty Stadium had prompted boss Paul Clement to call for a fast start.
But it was a false dawn as Swansea laboured for an opening and Brighton sensed this was an opportunity to climb the table further.
Pascal Gross saw Lukasz Fabianski tip his cross over and Alfie Mawson deflect his effort wide before Swansea's frustration was reflected by a reckless tackle from Federico Fernandez which left Murray in a heap.
Fernandez was maybe fortunate only to see yellow, but he was holding his head seconds later when he failed to react when Anthony Knockaert crossed from the right.
Murray stole in behind the stooping Argentinian to score for the third successive Premier League game, the ball deflecting off his knee past a helpless Fabianski.
Swansea seemed unable to clear their heads and Mawson missed Knockaert's throw to present Murray with another chance which deflected wide off Fernandez.
Fabianski came to Swansea's rescue from the resulting corner with a strong raised arm to deny Dunk's close-range header.
Swansea's 20 shots from their opening 10 games was the fourth lowest total in Premier League history and the first 45 minutes passed without another on target attempt.
The winger fed Abraham inside the area, but Seagulls goalkeeper Mathew Ryan was off his line quickly and spread himself well to block.
Swansea sent on Narsingh, Wayne Routledge and Oli McBurnie to provide further impetus, but Brighton held firm to move into eighth position.
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