The Jets, despite a number of odd-man rushes minutes earlier, had only one shot on goal at the time of Vrana's goal.
The shots on goal finished 10-2 Washington after the first, but the Jets led in attempts 13-12 at five-on-five.
"We just missed the net 10 times in the first period, so it looked like you had nothing going on," said head coach Paul Maurice.
"I thought we were a little bit tight with what we did. I liked where we ended up getting our shots from. You're over the net or you're a little late getting them off, a lot blocked. I thought we were a little bit tentative."
An Alexander Ovechkin interference call just under five minutes into the middle frame gave the Jets the opportunity to tie things up, and they wasted no time doing so. Scheifele won the ensuing offensive zone face-off and went to his spot in the slot. Quick passes from Dustin Byfuglien, to Blake Wheeler, to Kyle Connor at the side of the net opened some space, and when Connor's feed got to Scheifele, he made no mistake.
The assist moved Wheeler's point streak to 11 games, and it was also the third time on the home stand the Jets scored in the first 10 seconds of a power play.
"We have a lot of talented players. It was a great job all around there," said Scheifele. "Buff made a good play, Wheels made a good play, and KC made a good play. When they take away one option, we take advantage of the other one."