Hooker Richard Hibbard scored the only try of the match while Greig Laidlaw's six penalties kept Scotland in the type of match that Warburton relishes.
Wales' victory saw them pull level with Championship leaders England, who host Italy on Sunday, and maintained their hopes of defending their RBS 6 Nations title with the top two set to clash in Cardiff next Saturday.
But rather than bask in his own personal glory, Warburton, who ended the match as captain after Ryan Jones departed with a shoulder injury, chose to focus on the greater good.
"It's brilliant, I'm just so pleased we got the win and we're on a bit of a streak now, we knew it was going to be very hard coming to Murrayfield, it's always a battle," said Warburton.
"I didn't really feel I had a point to prove, it was just about getting straight back in and making sure that the team win again and we're very fortunate to have a fantastic squad and it's very competitive at the moment.
"But I'm just absolutely delighted to be back in the side."
For the third match running Wales prevented the opposition from scoring a try and Warburton believes defence coach Shaun Edwards deserves plenty of the credit.
He added: "Shaun emphasises the importance of defence every week, he's been fantastic, he's a superb defence coach and it worked out there, we wanted to stop Scotland and we managed to do that and we're very proud of that.
"We'll have 24 hours to relax and then we'll start looking forward to England on Monday."