Luka Milivojevic scored the game's opening goal from the spot after Federico Fernandez brought down Ruben Loftus-Cheek inside the penalty area.
Jordan Ayew then rescued a point for Swansea, firing a fine effort in from the edge of the box with 15 minutes to go.
Britton's appointment brought an improved performance from the Swans, but as has been so often this season their good play didn't correlate with three points as the home side struggled to create chances.
Less than a minute into the game and choruses of “Leon Britton’s barmy army” could be heard around the Liberty Stadium. Despite discontent among the fans, they were ready to back their club legend as he took charge of his first ever game.
The first chance of the game fell to Scott Dann, heading down from a corner that Lukasz Fabianski was gratefully able to hold.
Palace nearly took the lead through Andros Townsend, who cut inside and curled an effort from the edge of the box but Fabianski – the goalkeeper with the most saves in the Premier League this season - superbly tipped the shot over.
Swansea looked much more positive under Britton than Paul Clement this season, as Nathan Dyer saw a lot of the ball and was clearly instructed to run directly at Martin Kelly, but Palace were creating much better chances.
Loftus-Cheek flashed a shot wide as the away team looked very dangerous on the counter attack with the trident of the Chelsea loanee, Townsend and Wilfried Zaha.
Zaha then broke forward, beat Fernandez before he lost his footing in the box and squandered a good opportunity to break the deadlock.
Despite Swansea’s improvement, they failed to register a shot until the 65th minute. Failure to sign a Gylfi Sigurdsson replacement has cost them this season, as posters calling for chairman Huw Jenkins to resign were on display at the Liberty Stadium.
Familiar story for Swans
Despite not scoring a goal in their first seven league games this season, Palace have scored five more goals than Swansea this season and are five points above Britton's side.
The irony is that all that can save Swansea from dropping down a division is the January transfer window, as the club’s poor recruitment has been the biggest reason why the Swans have fallen from grace in recent years.
Palace take the lead
The first real action of the second half saw Palace awarded a penalty after Fernandez was adjudged to have brought down Loftus-Cheek inside the box.
There was no dispute over who would take the penalty today – with Christian Benteke suspended – as Milivojevic slotted the ball down the middle and past Fabianski.
The Eagles were on course to go unbeaten in their last eight Premier League games, a club-record as Swansea cranked up the pressure but failed to create any clear-cut chances.
Zaha and Townsend then broke away on the counter before the former curled a shot into Fabianski’s hands, showing that they will still pose a threat on the break even as they sat deeper following the goal.
Ayew the hero
Swansea needed someone to step up and be the hero, and Ayew was happy to fill that role.
The Ghanaian turned Milivojevic on the edge of the box before firing a fine strike that sent the Swansea fans into raptures.
Brimming with confidence, Ayew sent another strike inches wide just a few minutes later as Swansea began to play with more purpose.
The Swans pressed for a winning goal but didn't create enough to test Julian Speroni, as they picked up a bittersweet point.
Palace are an in-form side, and a point isn't a necessarily a bad result but Swansea still have the worst home record in the league, and need to start picking up wins soon.
Next up for the Swans is a Boxing Day trip to Anfield to face Liverpool. Whether Britton will be in charge remains to be seen, but problems lie deeper than the manager as they bid to avoid relegation for another year.