Higgins was behind 9-7 after the first two sessions of their second-round match but was out of the blocks swiftly on Saturday morning, winning the first two frames of the day.
It brought the scores level for the first time since the pair were locked at 2-2, but six-time champion Davis was not going to go quietly as he scrambled to win the next frame and go 10-9 ahead - needing just three more frames for victory.
That should have been two in the next frame, but on a 49 break he missed a simple red into the centre to hand Higgins a sniff. The Scot picked up a stunning green, coming off the cushion, but was unlucky when he finished smack up against the blue.
Three balls remained on the table and just two points the difference. Both players knew this was a key moment in the game with frames running out.
And when Higgins missed a long-range effort on the blue, Davis cut it in before coming back for the pink and re-opened his two-frame advantage.
A break of 59 rescued a frame back for the 34-year-old, one of only four players along with Davis, Stephen Hendry and Ronnie O'Sullivan to have won the famous Crucible trophy three times.
Maybe it was nerves, but both players were guilty of missing some straightforward pots in frame 22. Davis first missed a blue into the centre, then when we was gifted another chance, he ran out of position after sinking the black and his attempted cut on the red failed to drop.
Higgins finally got some table time and knocked in a 56 break to level the scores once more at 11-11.
With 12 reds remaining on the table, the next frame was a bout of safety heaven. One red was perilously close to the pocket, meaning neither player could allow to the white ball a line of sight. With the balls all spread, one error could cost the frame.
It took a superb long-range red from Higgins to break the deadlock, but he played safe up behind the green next to keep the match in limbo.
Davis failed to get on a colour as his break floundered at 26, Higgins then missed an easy black to hand Davis the initiative back at a tension-filled Crucible. The 52-year-old cleared the reds to stand 12-11 ahead, needing just one more frame for victory.
Higgins opened up with a 42 break in frame 24, but ran out of position in a frame he needed to win to stay in the competition.
Davis was next in the points, after an amazing miss on the red by Higgins showed the raw nerves from even these two experienced campaigners, and he held his nerve to clinch an amazing triumph.