It seemed Scotland would fall further behind when Murray trailed 6-3 but the former world number one won four of the next five points to level at 7-7 before a couple of match points came and went as Draper refused to wilt.
The two-time Wimbledon champion did not waste the third opportunity as a scorching running forehand down the line secured a 6-2 1-6 12-10 victory to the delight of those in attendance.
“(It was an) unbelievable ending, I’ve not played many tie-breaks like that before,” Murray told the BBC. “I wish (I could go on forever in tennis), unfortunately that’s not going to happen.
“I have put in a lot of hard work, it’s the hardest I’ve worked in a long time these last couple of months. I need to keep going because it’s not perfect yet but the signs are good. While I am still able to do it this is what it is all about.”
Murray is in action for Scotland all three days of this event, which pits the neighbouring nations against each other with bragging rights rather than ranking points on offer.
The three-time grand slam champion takes on Evans on Thursday before teaming up with brother Jamie, the tournament’s director, for a doubles match against Joe Salisbury and Neal Skupski to close the show.