Eichel tied it 1-1 on the same power play, shooting the puck over Koskinen's shoulder from below the left circle at 2:42.
Mittelstadt took a pass from Scott Wilson in the left circle at 13:29 to give the Sabres a 2-1 lead, and Eichel extended it to 3-1 at 17:24 when he looked off a pass on a 3-on-1 and scored blocker side.
"I thought [Koskinen], when it was 3-1, held us together," Oilers coach Ken Hitchcock said. "And then when we started to create some cycle game toward the end of the second, we felt a lot more comfortable in our game. But today was exactly what it was, it was a gutsy win, it was a never-say-die effort again. We're getting a lot of that now and we've just got to stay with it."