Selby led 4-0 and 5-2, but was forced to stagger over the finishing line with Davis refusing to buckle as the opening match of this year's tournament turned into a tight and sometimes turgid contest nobody could have predicted earlier in the afternoon.
Number one seed Selby appeared to lose his rhythm the longer the match progressed. He finally eked out an eminently forgettable final frame helped by Davis failing to escape from three snookers attempting to hit a red from behind the brown.
Selby has astonishingly won all nine of his final-frame deciders at the Masters.
He will face the winner of this evening's first round match between John Higgins and Stuart Bingham on Thursday night for a spot in the semi-finals.
"I think overall before the interval, I played quite well," said Selby. "I controlled the match to go 4-0 ahead, but I seemed to go back out and lose focus.
"For whatever reason, I struggled a bit. Every credit to Mark. He came back strongly. You just have to wipe out everything that has gone before and focus on the final frame.
"Just in general, I've had a lot of deciding frames and managed to come through more than I've lost. Beating (Ronnie) O'Sullivan here and stuff..it gives you confidence knowing you've won a few (final frames) in the past here."
Selby - Masters champion in 2008, 2010 and 2012 - collected the opening four frames without too much fuss aided by a 59 in the first frame, a knock of 80 in the second frame and a timely 58 in the fourth.
Davis had never been beyond the first round in two visits to this tournament. At one point, he failed to pot a ball for an hour or so as Selby cut a sharp figure before a sell-out 1500 crowd at the sport's most celebrated invitational event.
A humorous incident saw play delayed after the first and second frames to enable the crowd's earpieces to be replaced after unwanted background noise was found to be emanating from them in the arena.
It had little or no effect as Selby continued to express his will upon Davis, who was largely reduced to role of spectator in falling four frames behind.
Davis compiled 42 in the fourth frame, but it was to prove futile as Selby walloped a longish red into a baulk bag before clearing the colours.
Selby was first among the balls in the fifth frame, but an attempted cut on a red eluded him as Davis compiled 80 to ensure he would not suffer the ignominy of a whitewash.
He continued to unearth signs of a revival when Selby made 54 only to miss a yellow ball late in the sixth frame that would surely have handed him a 5-1 advantage.
With Davis leading 63-62, Selby produced a poor safety shot as his opponent rolled in a black to a centre pocket to cut the deficit to only 4-2.
Despite missing a red to a centre bag having compiled 35, Selby was not punished as Davis missed what looked like a straightforward red down a side cushion.
Selby added enough points to leave himself one frame short of the six frames needed to progress only for Davis to keep his hopes flickering with an imperious break of 136, a knock that sets an early target for the £10,000 highest break prize.
Selby was again forced to sit and suffer when he missed a long red late in the ninth frame. Davis steeled himself to contribute 49 to trail 5-4 including a wonderful positional shot to run the white around the angles to land on the black from the final pink.
A couple of loose safety shots encouraged Davis to piece together fine efforts of 62 and 61 to restore parity at 5-5 in forcing the 11th and final frame, but Selby held himself together to progress. Albeit in slightly unconvincing fashion.
"I never got a decent chance in the final frame," said Hastings man Davis. "But that's the way it goes. He held himself together in the end in the final frame because I was playing pretty well.
"It all came down to a few poor safety shots from me in that final frame."