A first-half stunner from Kevin Schade gave the Bees an early lead before Andersen levelled in the 76th minute.
After a dominant 3-0 win away at Fulham last week, Thomas Frank's men failed to put the game to bed which allowed the Eagles to claim a point in west London.
After a quiet start from both sides, Palace burst into life through the creative Eberechi Eze. The midfielder, recently linked with a move to Manchester City, hustled to win back the ball before feeding the marauding Odsonne Edouard who was dispossessed before testing Mark Flekken.
But it was the Bees who struck first in the 18th minute. Schade picked up the ball on the left hand side before driving at the Palace backline and producing a spectacular curled strike into Sam Johnstone's top-left corner, recording his first goal for the club.
Not long after, the heavens opened and the greasy surface favoured the elegant Mathias Jansen who sprayed the ball into attacking areas before a 38th-minute header from Yoane Wissa sailed over the bar.
Palace had looked flat all game and it was clear their only hope was through the dangerous Eze, who after 54 minutes slalomed his way through a wall of home defenders before being denied a penalty by referee Peter Bankes when he was brought down.
Soon after Edouard's dipping free-kick tested Flekken, who was forced to make a stretched save to deny the striker. And the Dutch keeper's heroics continued in the following sequence when he denied Jefferson Lerma's header from close range before Brentford scrambled to block Jordan Ayew's rebound.
Palace's persistence was rewarded 14 minutes from time. Andersen played a one-two with Ayew before driving into the box where he beat Flekken to the ball with his sliding effort.
Brentford searched for a winner during five minutes added time but the Bees struggled to find the final pass.