The British number two made a nervous start and lost the first three games but responded with a run of 11 in a row and eventually eased to a 6-3 6-1 6-2 victory on Court One.
Norrie, who was beaten by Rafael Nadal in the third round of both the Australian and French Opens this year, could now face eight-time Wimbledon champion Roger Federer, who was due to take on Richard Gasquet later.
Norrie was delighted with his performance, and said: “If there’s a decent time to play Roger it’s probably now but he still can play.”
Norrie joins Andy Murray and Dan Evans in the last 32, meaning three British men have reached this stage at Wimbledon for the first time this century.
The 25-year-old has been phenomenally consistent at a very high level in 2021, reaching three ATP Tour finals, most recently on grass at Queen’s Club, where he was beaten in three sets by Matteo Berrettini.
This was his 31st win of the year, putting him behind only Stefanos Tsitsipas and Andrey Rublev on the men’s tour, while, based on this year’s results alone, he is close to the top 10.
His ranking is actually at a career high of 34, enough to earn him a seeding at a grand slam for the first time, and he has certainly lived up to his billing.
He would have expected to beat 28-year-old Bolt, who has never made the top 100, but the Australian does know his way around a grass court and earned a wild card by winning one of the warm-up events in Nottingham.
Bolt made an impressive start, breaking Norrie with a fine pass and moving into a 3-0 lead. But from there it was one-way traffic, with the British player winning 18 of the next 21 games.
He won 11 in a row to move 5-0 ahead in the second, his relentless hitting, athleticism and mental approach simply too much for Bolt.
Bolt avoided a bagel but there was no way back and Norrie wrapped up victory in just an hour and 34 minutes.