The Eagles established a commanding 2-0 interval lead thanks to goals from Nick Carle and Darren Ambrose.
But Seasiders midfielder Charlie Adam struck shortly after the restart to haul his side back into contention and substitute Burgess drew the scores level with one minute remaining.
Blackpool boss Ian Holloway made a number of changes to his starting line-up after two straight defeats and on-loan Everton defender Seamus Coleman was handed his debut.
Ian Evatt, Stephen Dobbie and Barry Bannan also earned starting berths while Palace boss Paul Hart made three changes to the side which lost 1-0 at home to Leicester in midweek.
The Eagles caught Blackpool cold by racing into a third-minute lead.
A fine cross from the left flank by Lee Hills was met Carle, who fired a left-footed shot into the bottom-left corner from six yards out.
Blackpool almost fashioned a quick response when a free-kick by Bannan was greeted by Alex Baptiste but his leader from 12 yards flew over the crossbar.
In the 14th minute, Dobbie collected possession and engineered the space to hit a 20-yard strike which Clint Hill did well to block.
Moments later, an outswinging corner from Ambrose was met by Hill but his header flashed just wide from six yards out.
However, Blackpool continued to probe and a shot by Dobbie narrowly missed the target from 20 yards in the 28th minute.
Holloway's men went close again on the half-hour mark when a cross from Coleman was met by Bannan but his header flew wide.
However, four minutes later Palace established a two-goal cushion when Ambrose claimed his 18th goal of the season.
The goal came after a long throw-in from Danny Butterfield and saw Ambrose apply a clinical right-foot shot from just six yards out.
Palace went in search of a third goal in the 36th minute when Alan Lee fired goalwards from 20 yards but Baptiste made a fine block.
Hill then tried his luck but Baptiste was again on hand to avert the danger.
Holloway made a tactical switch in the 37th minute, replacing Stephen Husband with Burgess up front in a bid to find a more physical presence.
And moments after the restart, Blackpool were right back in the game when Adam collected a pass from Dobbie and drilled home a left-footed shot from 20 yards.
He tried his luck from distance again moments later as the Seasiders begin to up the tempo.
In the 57th minute, Blackpool carved out a fine opening when Coleman crossed dangerously and Billy Clarke's header was well saved by Julian Speroni.
Dobbie then fired wide for Blackpool as the Eagles mounted a spirited defensive effort to keep the hosts at bay.
Adam and Gary Taylor-Fletcher both went close for the Seasiders before Burgess fired home Ian Evatt's pass with a left-footed shot from six yards.