Tom James’ 96th-minute winner secured all three points for the green and whites who had been on a 12-match unbeaten run against John Sheridan’s side.
Jon Obika and Dean Bowditch were on target seven years ago at Brunton Park, with James’ late free-kick moving Town up to 12th and within six points of the play-off spots.
Speaking to iFollow Glovers, Way was impressed with elements of his side’s display.
“The effort that was shown, and the application, and I thought at times we were back to our best.
“We talked about it before the game. Making sure we keep calm under pressure. I thought the way we moved the ball, we used the pitch – it’s the first time we’ve used this width of the pitch, and players just kept passing and passing and passing.
“The players kept going and that’s what pleased me more, that they were very very brave to come to Carlisle and play the way that we did.”
Yeovil faced the longest trip of the weekend in the EFL, with a 660-mile round trip.
Rewarding travelling supporters with a pie and a pint at Brunton Park, Town’s boss was pleased to provide all three points, too.
“I really appreciate the supporters that have stuck behind these players, because they are young players.
“That’s the way to show supporters that they really care for the football club. You could see the emotions after.
There were one or two of our supporters there where you could see they needed that. They sung throughout the game, and hopefully they had a safe journey home.”