James Scowcroft and skipper Shaun Derry both saw red for two bookings in the opening seven minutes of the second half, to put the home side's backs firmly against the wall.
But Neil Warnock's resilient side held on for a point, which typifies the grit and determination embodied by their Yorkshire manager.
In an opening period lacking in any real quality, Palace created the first opportunity in the 15th minute when John Oster forced Brian Jensen into a comfortable save from a 20-yard free-kick.
The Clarets created their first chance in the 35th minute, but Martin Paterson's curling effort was easily saved by Julian Speroni, before Scowcroft received his first yellow card from referee Mr Williamson, for what seemed a deliberate elbow on Mike Duff a minute later.
If the opening period was void of any talking point, the second half was the complete opposite. Just two minutes after the restart Scowcroft was given his marching orders for another elbow, this time on Clarke Carlisle.
Remarkably, just five minutes later, Eagles captain Derry was shown a red card for two separate offences, the first being for dissent and the second for a poor challenge on Wade Elliott.
Kevin McDonald, Christian Kalvenes, Chris Eagles and Paterson all went close to giving Burnley the lead, before Chris McCann finally had the ball in the net after bundling home substitute Robby Blade's 75th-minute corner, only for the controversial referee to rule out the effort for a foul on Speroni.
Graham Alexander saw his spectacular long-range volley well saved by Speroni in stoppage time and somehow Palace held on for an unlikely point as both sides continued their winless Championship start.