The Bruins got the power play in the second extra period when Buffalo's Drew Stafford was caught on the ice as a sixth skater. Satan got a pass from Michael Ryder in front of the net and put it past Ryan Miller.
Tuukka Rask made 35 saves for Boston and also assisted on Satan's goal.
Ryan Miller stopped 36 shots for the Sabres.
Game 5 is Friday night in Buffalo.
Tim Kennedy and Steve Montador scored for Buffalo, which opened a 2-0 lead after two periods. David Krejci and Patrice Bergeron scored in the third period for Boston to send the game into overtime.
Both goalies made stunning saves to keep their teams in the game.
Rask reached out to knock away a puck that even 6-foot-9 defenseman Zdeno Chara couldn't reach, preserving the tie with 11 minutes left in regulation. Miller stopped Blake Wheeler on a short-handed breakaway, then got a glove on a sneaky backhander from Satan early in the first overtime.
Miller surprised even the scoreboard operator, who sounded the goal horn briefly.
But they were off to another overtime.
In the second extra period, Miller took a shot from Milan Lucic off the chest protector and doubled over in pain. Just 31 seconds later, the Sabres were caught with too many men on the ice, and though Stafford did a quick about-face to scramble back to the bench, it was too late.
For the fourth straight game, the Sabres scored first, going up 1-0 just 2:12 into the first period when Wheeler was called for a delayed penalty; Kennedy scored his first career playoff goal before Miller could even get off the ice for an extra attacker. They made it 2-0 with 7 minutes gone in the second when Montador's slap shot from the point was deflected by traffic in front of the net past Rask.
The Sabres were 30-0 in the regular season when leading after two periods, but Boston snapped the streak in Game 2 when it rallied from a 3-2 deficit with three third-period goals to win and swipe away home-ice advantage. And the Bruins were did it again in Game 4.
With about 2 minutes gone in the third, Cody McCormick slammed into Rask trying to outrace a defender for a puck and was sent off for goalie interference. Eight seconds into the power play, Krejci picked up a rebound in the slot, moved himself to the right and shot it past Miller before he could get back into position.
With 13:20 left in regulation, Bergeron came up with the puck to Miller's right and fired it past him, just inside the far post. Buffalo called a timeout and settled down, preserving the tie into overtime.
Mike Grier nearly gave Buffalo back the lead when he shot an an open side of the net — past Chara — with 11 minutes left in regulation. But Rask, diving behind his defensman, got his blocker on the puck to knock it away.
Notes
The Bruins honored the NCAA championship team from Boston College during a first-period break. BC won its fourth title with a 5-0 victory over Wisconsin on April 10. … McCormick, recalled from Portland of the AHL on Tuesday, assisted on the Sabres' first goal. … Buffalo's Thomas Vanek missed his second straight game with a high ankle sprain. He was seen in the elevator wearing a boot before Game 4. … The Sabres were 0-for-2 on the power play, and are 0-for-14 with an extra man in the series.