Norrie had coasted through the first set, recovering from losing the opening service game to break Nishioka three times.
However, Nishioka, the world number 66, rallied in the second set, with Norrie guilty of some unforced errors after losing his serve at 2-2 and again to trail 5-3.
Norrie, ranked 56 in the world, was soon up against it in the decider as Nishioka broke twice to take a commanding 5-1 lead and seemed on course for victory.
However, the 25-year-old then produced his own fightback to level the set at 5-5 with another break before clinching a sixth straight game, once more against the Nishioka serve, to progress in two hours and 16 minutes.
Norrie will play ninth seed Grigor Dimitrov in the second round.
Katie Boulter produced a composed display to book her place in the second round of the WTA tournament after coming from behind to beat Czech player Kristyna Pliskova 4-6 6-3 6-3.
Both women struggled to hold onto their service games in the opening set, which the world number 78 eventually closed out.
Boulter, who went out in the first round at the Abierto GNP Seguros in Monterrey, started a fightback to move 4-1 ahead with a break midway through the second set and forced a decider.
The 24-year-old continued her momentum with an early break and hold to lead 3-1, before eventually taking a fourth match point as Pliskova’s serve again failed to stand up following a run of double faults.
Boulter will next face Belgium’s world number 17 Elise Mertens.
Heather Watson will not join Boulter in the second round, after she was knocked out in straight sets by Serbia’s Nina Stojanovic.
The 65th-ranked Brit was left to rue some costly early errors as she dropped the first set 6-4 before Stojanovic blew her off the court 6-1 in the second.
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