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In a repeat of the 2015 quarter-final, McGill gained revenge despite falling 5-4 behind after the first session.
A high-quality encounter saw world number 30 McGill and 2005 champion Murphy make two centuries each.
Five-time champion Ronnie O'Sullivan opened up a 6-3 lead over qualifier David Gilbert in the first round.
O'Sullivan made breaks of 82, 74 and 71, but world number 22 Gilbert showed his ability with a superb 125 clearance. They play to a finish on Monday from 10:00 BST.
Alan McManus progressed with a 10-7 victory over fellow Scot Stephen Maguire, who lost in the first round for the fourth consecutive year. McManus will face Ali Carter in the second round on Thursday.
In Sunday's evening session, Welshman Mark Williams took a 7-2 lead over fellow former champion Graeme Dott of Scotland, while England's Sam Baird led Michael White of Wales 5-4. Both matches resume on Monday (19:00 BST).
Murphy, 33, becomes the third world champion after Peter Ebdon and Stuart Bingham to exit the competition.
This is only the second time that the previous year's finalists have both fallen at the first hurdle.
The last occasion was in 1980 when Terry Griffiths and Dennis Taylor both lost in the second round after receiving first-round byes.
McGill, 25, comes up against Ebdon's conqueror Marco Fu in the next round.
"I'm obviously disappointed but having a child on the way has change my perspective on things," said Murphy. "It is just a game of snooker.
"My preparation could not have been better. My game was in better shape than last year.
"If Anthony can keep his composure, keep himself under wraps and play with enjoyment, he will take some stopping."
McGill made his debut at the event last year, beating Stephen Maguire and ousting then defending champion Mark Selby, before falling to Murphy in the last eight.
Despite making only his second appearance at the Crucible, the Scot showed no sign of nerves and looked comfortable around the table.
His ruthlessness among the balls - managing breaks of 119, 117, 97 and 85 - underlined his quality.
"When I came here last year, I had no ambitions of winning and I just wanted to enjoy it," said McGill.
"It is an honour for me to play here and a lot of people around the world want to do it. It is a privilege."

EJ has weathered a snow storm and harassment from rival ospreys while incubating her clutch of three eggs. All hatched over the past few days.
The RSPB suspects mate Odin, who was bringing fish to the chicks, was frightened off by another male.
On Monday, an osprey was at the nest with a fish but the chicks were dead.
The bird is thought to have been EJ.
RSPB Scotland said she was expected to remain at the nest and guard it against being taken over by other females.
It is possible to view the nest site via a webcam. Footage on Monday morning appeared to show the nest had been abandoned.
Later, on occasions, an adult osprey was seen at the nest, at one point with a fish.
At the weekend, RSPB Scotland said it had decided not to intervene in what it described "heart breaking" but natural process playing out at its reserve.
EJ, who is 20 years old this year, has been visiting Loch Garten for 15 years and has reared 25 chicks over that time with mate Odin and other males.
The nest is in a tree on an RSPB Scotland reserve.
EJ and her long-term mate Odin are the most successful breeding pair at the Loch Garten site.
Over previous seasons, 17 of their chicks have fledged.
Ospreys migrate from west Africa to Scotland to breed and can be seen hunting for fish from rivers and lochs.
EJ's breeding seasons have been fraught with drama in the past.
Incidents have included eggs being kicked out of the nest by a male bird and battles between EJs and rival females over the nest.
Last year's finalist Shaun Murphy suffered a shock 10-8 first-round loss to qualifier Anthony McGill at the World Championship in Sheffield.
[NEXT_CONCEPT]
All three chicks being raised by a well-known female osprey in the Cairngorms have died, RSPB Scotland has confirmed.