Clement was appointed as Bob Bradley’s successor earlier on Tuesday and was at Selhurst Park to watch from the stands as his side took on Sam Allardyce’s Eagles.
And it turned into a dream start as Alfie Mawson headed them in front just before the interval, before Angel Rangel scored a late winner just after Wilfried Zaha’s spectacular strike seemed to have secured his side a share of the spoils.
The win moves the Swans above Hull and level on 15 points with Sunderland – just a point behind Palace in the final position of safety.
Swansea started brightly and went close to going in front in the sixth minute when a Wayne Routledge ball into the area picked out Fernando Llorente but the big Spanish hitman could only send his header wide of the goal.
Both sides had an opportunity to go in front within the space of a minute as Yohan Cabaye’s 16th-minute free kick was cleared back to him and his follow-up strike from a tight angle was saved by Lukasz Fabianski.
The Swans broke quickly and Llorente picked out the run of Jack Cork inside the area and his eventual strike was well saved by Wayne Hennessey.
From the subsequent corner Gylfi Sigurdsson picked out Ki Sung-Yeung but he somehow failed to hit the target in the area.
Swansea continued to look the more likely to score and went close again on 27 minutes when Sigurdsson’s free-kick into the box was met by the head of Federico Fernandez who failed to keep his effort down as it sailed over the bar.
On 32 minutes Routledge went through on goal but failed to get a clean connection on the shot.
A minute later and Palace had a good shout for a penalty when Christian Benteke seemed to be clipped by Fabianski but referee Paul Tierney waved play on.
Swansea got the goal their play deserved on 42 minutes when another Sigurdsson free kick was met by Mawson in the area and he beat Hennessey with a flick header into the corner of the net.
Swansea started the second half brightly and Sigurdsson was again the architect of another chance but Cork could only fire over the top from the edge of the area on 57 minutes.
Two minutes later and Palace substitute Bakary Sako tried to catch out Fabianski with a free kick which was heading just inside the near post until the Poland international moved quickly to push it over.
Palace continued to up the tempo and Joe Ledley will no doubt have thought that he had fired them level on 63 minutes but his effort was blocked by team-mate Fraizer Campbell who was laid on the floor at the time.
Swansea were pushed deeper into their own half as the half wore on and survived another penalty shout on 77 minutes when Sako went down under a challenge from Rangel but again the appeals were ignored.
Palace’s pressure paid off with an equaliser on 83 minutes when Martin Kelly’s cross appeared to be going behind Zaha, but the winger produced an excellent scissor-kick volley past Fabianski.
However, Swansea secured a massive win courtesy of Rangel’s 88th-minute strike when he collected a ball over the top from Leroy Fer and produced a composed left-foot finish past Hennessey