Heta came from 2-0 down to tie the match up but he missed six match darts to complete the comeback and 'The Gambler' moved through to a tie with 'Jackpot' Adrian Lewis.
Baggish produced a fine start to proceedings taking out 113 in the deciding leg to claim the first set.
It was all going the way of 'The Gambler' against the in-form Aussie as he went two sets up in quick order with a 76 checkout in another decider to double his lead.
But this is when it got interesting for Heta as he took out 146, 81 and 160 to claim the third set 3-1 and stay alive in proceedings.
It was sent all the way to a decider with Heta following that up with a 100 checkout to claim a clean sweep and make it 2-2.
Another turn was set as it was Heta's time to hand Baggish the initiative in the form of five missed match darts and Baggish pinned double one to send it the distance at 2-2.
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This proved pivotal as despite surviving another match dart, Baggish hit double 20 to move through.
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𝗕𝗔𝗚𝗚𝗜𝗦𝗛 𝗦𝗛𝗢𝗖𝗞𝗦 𝗛𝗘𝗧𝗔
Danny Baggish stuns Damon Heta in an absolute thriller here at the Palace.
Nearly an amazing comeback from the Australian but he missed six darts for the match and crashes out!