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After erasing the Sabres' 3-1 lead with three goals in the final 3:31 of the middle frame, the Oilers caught some luck in the third when Jason Pominville stopped his own attempt from going in with the follow-through of his shot as the cage yawned wide open.
"It was right until the end," said Brodziak. "There were crazy bounces and a double hit on an open net that stayed out. There was a lot of chaos.
"It almost had a playoff-like feel to it."
There's been no hotter stick in the National Hockey League than Draisaitl's and the German kept it that way with his 17th of the year 1:28 into the game.
The Deutschland Dangler deposited his 41st of the season shorthanded, ripping in a one-timer while on a 2-on-1 with McDavid.
Draisaitl's goal marked the second time in his career he's achieved an 11-game point streak.
"Obviously I've been feeling good and hopefully I can keep it up," said the forward of his performance of late, elevating his point total to 83.
The Oilers lead was shortlived, as the Sabres responded three times unanswered, with 2015 second-overall selection Jack Eichel scoring twice.
Buffalo's captain replied on the power play just 1:14 after Draisaitl's goal, lifting the puck over Koskinen's shoulder for the equalizer.
"We looked like a team that was on the end of a long road trip at the start," said Oilers Head Coach Ken Hitchcock. "We got some juice in the second period and then a couple breaks in the third."
Casey Mittelstadt added to the lead at 13:29, whipping one past Koskinen from the left circle for his 10th of the 2018-19 season.
Down 2-1, Oilers forward Zack Kassian fired a shot off the rush that went wide and Eichel turned up ice in transition. The Massachusetts product elected to keep, shoot and score on a 3-on-1, beating Koskinen blocker side to give the Sabres a 3-1 lead into the middle stanza.
"We did an unreal job to come back and tie it at 3-3," said Brodziak. "We got a deflection that was a bit of a lucky bounce but we'll take it."