Burrow, Australia's number one surfer, lost his third-round encounter with compatriot Kai Otton at the ASP World Tour event in Victoria.
Otton had previously beaten Julian Wilson to make it to the third round.
Slater, the 11-times world champion, scraped through his heat after being given an almighty scare by Australian wildcard Nic Muscroft.
The American, winner at Bells Beach on four occasions, grabbed the final wave with eight second remaining to post a 9.23 score and secure a late victory over Muscroft.
Victory for Slater means his tilt to become the first man to win the Rip Curl Pro five times remains on. The only other surfer to have won it four times is Mark Richards.
"You go through the whole week, you dodge bullets, you make choices good or bad. If you come on top (at Bells), you clearly should have been the guy who should have won," Slater said.
"I've got four, but it took me 20 years to do that.
"Today was a full battle back and forth. He was always in it. Somehow I was lucky to have eight seconds left and get that one."
Defending champions Joel Parkinson will join Slater in the next round after beating beat Hawaiian Fredrick Patacchia.
Parkinson, himself a three-times winner at Bells, delivered a stunning equal event-high 9.87 to move through in solid one to 1.5-metre waves.
Fellow Australians Mick Fanning, Owen Wright and Josh Kerr also progressed on day three of action with the fourth round set to start later on Thursday.