Palace midfielder Darren Ambrose blasted a third-minute penalty over the crossbar after Moses had been fouled by Barnsley defender Ryan Shotton, but a point each was a fair reflection of this mid-table tussle.
Barnsley were chasing a third straight league win, but extended their unbeaten run to seven matches, while Palace's topsy-turvy season continues after they bounced back from last week's defeat at manager Neil Warnock's former club, Sheffield United.
Palace spurned a golden chance to take an early lead after Shotton had been penalised by Hertfordshire referee Keith Hill for his challenge on Moses in the penalty area.
Ambrose stepped forward, but blazed the subsequent spot-kick, right-footed, over the crossbar.
Palace gave their Yorkshire visitors another big scare 10 minutes later. Ambrose swung over a corner and striker Alan Lee's goalbound header was brilliantly turned round the post by Barnsley goalkeeper Luke Steele.
But Barnsley responded soon after and were much more ruthless with their first chance. Anderson De Silva robbed Palace defender Nathaniel Clyne down the right and his cross to the edge of the penalty area was expertly curled into the bottom corner by Bogdanovic in the 19th minute.
Steele pulled off a second fine save in the 24th minute when keeping out a stinging 20-yard volley from Moses, who was prominent throughout, as Palace were held at bay until half-time.
Barnsley manager Mark Robins sent Iain Hume on for goalscorer Bogdanovic three minutes after the restart, but Moses picked up where he had left off in the first period and soon demonstrated why he is one of the Championship's hottest talents.
The Eagles pushed forward down the left and from Lee's headed pass on the edge of the area Moses powered a spectacular bicycle kick into the top corner for a 51st-minute equaliser.
Palace defender Claude Davis then rifled a left-footed volley over the crossbar when he should have at least been on target, but the visitors showed plenty of signs of their recent improvement and Andy Gray fired a 25-yard shot straight at home goalkeeper Julian Speroni.
Both sides went in search of the winner in the last 20 minutes. Ambrose lashed a 20-yard free-kick over the bar and then crossed for defender Clint Hill to force another save from Steele with a firm header.
But Tykes substitute Adam Hammill arrowed a right-footed shot from the edge of the area just wide and Iceland international Emil Hallfredsson would have snatched it for the visitors from 25 yards but for Speroni's flying save.