After the teams exchanged goals in the opening 45 minutes, with striker Glenn Murray cancelling out Adam Bedell's opener, Palace seemed on track for victory when second-half substitute Jake Gray bundled the ball over the line in the 70th minute.
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Palace - contesting the first of three fixtures in the United States - arrived in the US on the back of a 14-1 rout of seventh-tier Austrian club Grazer AK last week.
But coming up against a Columbus outfit in the midst of a MLS campaign, Tony Pulis' men found the going tough.
Columbus came closest to opening the scoring through Agustin Viana in the 10th minute, though the defender's header cannoned off the post and into the hands of goalkeeper Julian Speroni.
The Crew were rewarded for their efforts four minutes later, when towering striker Bedell - in his first year at the club - guided his header across goal and into the far left corner.
Palace levelled proceedings 10 minutes before the half-time interval courtesy of Murray, whose glancing header left Crew goalkeeper Matt Lampson with no chance.
Pulis made a host of changes after the break, with captain Mile Jedinak, scorer Murray and shot-stopper Speroni making way for Marouane Chamakh, Barry Bannan and Wayne Hennessey.
Chamakh made an impact in the second half, hitting the post in the 70th minute before fellow substitute and youth-team star Gray pounced on the rebound.
Not to be outdone, Columbus restored parity 10 minutes later thanks to new recruit Paladini and subsequently held on to earn a well-deserved draw.