The Argentine shot-stopper flung himself to his right to keep out the midfielder's spot-kick in the 85th minute after Lukas Jutkiewicz had been hauled to the ground by Mile Jedinak.
The result ensures both sides continue to search for their first league win of the season, while Burnley's afternoon was further soured by last season's top scorer Danny Ings hobbling off shortly before half-time with a hamstring injury.
Sean Dyche's men now look set to be without both leading strikers from last season as Sam Vokes continues his recovery from a serious knee problem, and they were unable to carve out any clear goalscoring opportunities until Jedinak's foul offered them the chance of taking all three points back to Lancashire.
Palace had had the better of the chances up until that point, and almost had an early lead when Scott Dann struck the crossbar in the opening stages, but the hosts were unable to find a way through in Neil Warnock's first game back in charge at Selhurst Park.
After shipping three goals in the draw at Newcastle United, Warnock will at least have been pleased to see his side secure a clean sheet, with Palace's Premier League survival last season built upon a strong defensive record under Tony Pulis.
Warnock made three changes from the game at St James' Park, including a debut for record signing James McArthur in midfield and a first start for Wilfried Zaha in his second spell back at the club.
Burnley showed just one alteration from their stalemate with Manchester United before the international break, as new signing George Boyd made his bow in place of Matt Taylor on the left flank.
The hosts twice went close to taking the lead inside the opening eight minutes as Damien Delaney's shot was deflected narrowly over the angle of post and bar before Dann headed Fraizer Campbell's delivery against the woodwork.
Zaha then thumped a shot straight at Tom Heaton, with Burnley captain Jason Shackell forced to clear off the line after the ball had rebounded off his shin.
Midway through the half, Dwight Gayle snatched at a chance and fired over following a poor header from Michael Duff.
Palace began the second period as they had the first and Gayle was agonisingly close to getting a telling touch on Mile Jedinak's header across goal in the 51st minute.
McArthur fired a volley narrowly over the bar soon afterwards, while Arfield tested Speroni for the first time with a stinging drive from the edge of the area.
Burnley looked set to claim a maximum return when referee Mike Dean pointed to the spot after Jedinak's misdemeanour, but Speroni came to his captain's rescue as both sides were forced to settle for a draw.
