The British number one was faced with a must-win match after Dan Evans was beaten 6-4 4-6 6-4 by Tommy Paul in a high-quality opening rubber at the Emirates Arena.
Norrie made a very unpromising start, quickly losing the first set, and, after seeing an early break disappear, was hanging on for much of the second set.
But he managed to force a tie-break, seized his chance in that, and then served out a 2-6 7-6 (2) 7-5 victory at the second attempt just after 9.30pm to the delight of a packed crowd.
The result means the tie, Britain’s opening contest of a four-team round-robin group, will be decided by the doubles rubber, set to feature Andy Murray and Joe Salisbury against Salisbury’s US Open-winning partner Rajeev Ram and Jack Sock.
Norrie, whose Glasgow-raised father David was among the crowd, said: “Honestly I owe it all to the crowd, you guys were amazing. I wasn’t playing my best tennis, Taylor came out firing.
“I managed to hang in there in the second set, I played well in the tie-break and hung tough in the third set again. I loved the atmosphere, it’s just great to be back in Glasgow.
“I knew what I had to do, I had to come out and get a win for the team and I managed to do that.”
A sombre start to the tie reflected the period of national mourning following the Queen’s death, with the usual music replaced by a lone piper and a minute of silence before the national anthems, while the British players and support team wore black ribbons.
Music has been very much a part of the great Davis Cup atmospheres for previous matches at the Emirates Arena, including a crucial win over the USA to kick off Britain’s run to the title in 2015, and Evans felt the difference.
He said: “It was sombre at change of ends. Something was missing. That’s what it is at the minute.
“We’re very lucky to be playing. Thankfully the event was allowed to go on. We’re just doing the best out of what we can. It was still a good atmosphere, I still enjoyed it.”
Evans was also ranked four places higher than his opponent but Paul is a player with considerable all-round quality and in form after a strong summer.
There was little to choose between the pair throughout an entertaining contest full of lung-busting rallies.
Evans was twice a break up in the opening set but was under pressure in every service game and Paul responded straight away on both occasions before clinching his fourth set point.
Evans hit back well in the second and gave himself a chance by breaking Paul back when he served for the match at 5-3 only to drop serve again himself.
The 32-year-old said: “It was a good match. I probably lost the match in the first set. Very slow court. It took a lot to get used to the balls.
“All credit to Tommy. I thought he served well when I had my chances. It’s a tough one. But another good effort I guess. That’s Davis Cup, isn’t it? We knew this one was going to be tough. It certainly proved to be for me. I played good tennis. Just couldn’t get over the line.”
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