The 44-year-old former Bayern Munich assistant, whose appointment was confirmed hours before kickoff, began the game in the stands but ended it on the touchline as Angel Rangel fired an 88th-minute winner past Wayne Hennessey.
Swansea were the better side overall and took the lead through an Alfie Mawson header, his first Premier League goal, from Gylfi Sigurdsson's cross just before halftime.
Palace improved in the second half and thought they had secured a point when Wilfred Zaha, playing his final game before heading off to the African Cup of Nations with the Ivory Coast, met Martin Kelly's cross with a spectacular scissor kick on 83 minutes.
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But Rangel made the difference to inflict defeat on Palace boss Sam Allardyce in his first home match in charge.
Swansea caretaker coach Alan Curtis took charge of the team for most of the game but hailed the impact of the incoming Clement.
"Paul came down and made a real positive contribution. He came into the dressing room before the game and at halftime and at the end and played a part in the substitutes," Curtis told reporters.
"There's nothing better for any player than a new manager in the stands looking at you, and that's the same for the players on the bench and those back in Swansea. When you've got a new manager in place it gives everyone a huge lift. It is a terrific result and a huge three points."
Allardyce received a warm reception in his first game in the Selhurst Park dugout but the home fans turned on his players after a lifeless first-half display, which the former England manager blamed on the fact his team had had one day fewer to prepare than their opponents.
"It's easily identified, the lack of energy the players showed. We struggled to keep up with Swansea, we hadn't recovered properly," said Allardyce, whose side hover one point above the relegation zone in 17th.
"You can see it with your own eyes, you don't need to be a football manager. Some people say it's rubbish but it's not, the players were trying 100 percent but they were not physically able to reach their usual levels. They are shattered," he added.
(Reporting by Neil Robinson; Editing by Toby Davis)
