The Manchester United star made it seven in his last seven internationals with two second-half efforts and is now just four goals short of becoming England's all-time top goalscorer - currently sitting in third on 46 behind Gary Lineker and Sir Bobby Charlton.
Roy Hodgson's visitors led through Alex Oxlade-Chamberlain's neat header and looked to have the game sewn up early in the second half when Rooney nodded home a second.
Scotland briefly threatened a comeback when Andy Robertson rounded off some slick passing involving substitute Johnny Russell but Rooney slammed home his second and England's third with six minutes remaining to silence the home support once again.
The 29-year-old told Sky Sports afterwards: "It's a great victory for us tonight. We spoke about the atmosphere before the game. It's a terrific atmosphere here at Celtic Park and we had to show our character here tonight.
"It was an amazing atmosphere. You could hear it in the national anthems. We showed our ability with the ball and our mental strength tonight. I thought after we scored the first goal, we controlled the game.
"When Scotland got their goal back, it could've been edgy but we responded fantastically well so it's a great victory and a great performance.
"I think it's a great fixture for the fans and a fantastic fixture to play in. It's obviously not my decision but it'd be great to have more games like this."
When asked for his feelings on creeping closer to the all-time England goals record, Rooney added: "I'm just happy to score goals for England. It's an unbelievable feeling. Tonight was very special for me, to score two goals for England at Celtic Park against Scotland, it's a special night for me tonight."
Peter Shilton's record of 125 England caps is also within reach and Hodgson has little doubt the player will achieve both milestones sooner rather than later.
"Records and milestones are important," the England coach said. "It must be nice for him to think he'll see his name at the top of the list, and a record number of caps is in his sight too.
"He must keep his fitness, but he won't be the first one to retire from football.
"He did terrifically well tonight, but many players took responsibility out there."
