Burroughs' win clinched the third-seeded Indians a date into Tuesday evening's semifinal against second-seeded South Torrance (22-4).
At two different points, the Indians (21-4) were nearly pushed to a fourth game.
Burroughs, which hadn't trailed at any point until the third game, fell behind, 21-19, and then, 23-20, in the third game.
The Indians responded to the second deficit with a 3-0 spurt culminated on an ace from Daniel Marbach quickly followed up by a service error that propelled the Spartans (21-9) to game point at 24-23.
Yet, Burroughs responded with consecutive kills from Daniel Jacobs (team-high 10 kills) in rushing to match-point at 25-24.
Rio Mesa answered back with a point and both teams traded the lead until the Indians went ahead for good on a kill from Marbach at 28-27.
The spike was followed by a stuff from middle blocker Ike Nwachie, the final of his team-best four, which secured the win.
"We're our biggest enemy," Jacobs said. "We didn't do well and we know we're going to need to play better in the next round."
While Burroughs never trailed in the first game, the Indians found themselves even at 24 before they again surged ahead on a kill from Jacobs.
Again, though, Burroughs had difficulty putting away Rio Mesa, but was inspired by its defense when a sprawling pancake dig from defensive specialist Danny Casillas turned into a point when the ball dropped between two Spartans on the other side of the net.
On the next possession, Robby Rutecki (eight kills and nine digs) iced the game with a kill through the block attempt of Sean Bisnett.
"The defense was going to do whatever it could to get us going," said libero Derek Nicandro, who led the team with 15 digs. "We didn't play great, but the goal was to get to the next round."