It was a contest which ebbed and flowed for three-and-a-half hours before being suspended at the start of the fourth set on court eight and later moved under the roof at the Margaret Court Arena.
Norrie had not made it out of the first round at the Australian Open in two previous attempts but the result set up a third-round encounter against Rafael Nadal.
Nadal showed no signs of the back issue which hampered his build-up in Melbourne in a comfortable 6-1, 6-4, 6-2 victory over American Michael Mmoh.
But play was briefly halted in a bizarre moment when Nadal was serving for the second set only to be harangued by a woman in the stands.
The woman, who was later escorted out by security, shouted: “Hurry up, you OCD f**k” as Nadal prepared to serve before raising a middle finger towards the Spaniard. Nadal replied by saying “what me?” and smiling.
It failed to put him off his stride in a comfortable win, as Norrie also backed up his first-round triumph against fellow Briton Dan Evans with victory over his Russian opponent.
Afterwards, the British No1 said he was given a lift by the move to a show court.
He said: “I definitely used the crowd tonight. I’m playing Rafa next and that’ll be on big court so that’s good.”