Abe was born in Tokyo to Mitsuo Abe, a paved flat track rider. He began racing minibikes at the age of eleven and spent his early competitive years in motocross. At fifteen he switched to road racing and also competed in the United States. In 1992 he was runner-up in the 250cc category of Japan's domestic National A championship. In 1993, competing in the All Japan Road Race Championship, Abe won the 500cc title in the category's final year, becoming the youngest title winner in that class.
In 1994, while still racing in his home championship, Abe was given the chance to enter the Japanese Grand Prix as a wild card. His performance was extraordinary: he challenged for the race win in the premier 500cc class until falling off three laps from the finish. The ride impressed Kenny Roberts's Yamaha team sufficiently that Abe was offered two further rides that year, returning sixth place finishes on both occasions, and earning him a full-time Grand Prix seat for 1995.
The impact of that 1994 wild card appearance extended beyond Abe's own career. A fourteen-year-old Valentino Rossi, watching the race, was so impressed by Abe's commitment and spectacle that he adopted the nickname "Rossifumi" — a portmanteau combining his own name with Abe's given name Norifumi — and used it in his early career as a tribute.
Abe took his first podium in 1995, followed by his first Grand Prix win in 1996, when he also finished fifth overall in the championship. His 1997 team was managed by former champion Wayne Rainey, and Abe recorded four podiums across the next two seasons while finishing consistently in the points.
In 1999, having joined the d'Antin Antena 3 team, Abe won at Rio de Janeiro. He followed that with another victory at Suzuka in 2000. Despite spending two subsequent seasons on less competitive machinery, Abe maintained a perfect record of top-ten championship finishes throughout his 500cc Grand Prix career.
The transition to MotoGP regulations in 2002 brought four-stroke machinery that did not suit Abe's style. When the d'Antin team switched to the Yamaha YZR-M1 for 2003, Abe left to serve as a factory test rider and occasional wild card racer for Yamaha. He returned to a full-time seat with the Tech3 Yamaha team in 2004 but the results did not come, and he moved to Yamaha's Superbike World Championship squad for 2005 and 2006 without managing a podium in either season.
In 2007, Abe returned to Japan to race in the All Japan Superbike Championship on a Yamaha.
On October 7, 2007, Abe was riding a 500cc Yamaha T-Max scooter in Kawasaki, Kanagawa, when a truck made an illegal U-turn in front of him. The collision occurred at 6:20 p.m. local time. Abe was taken to hospital but was pronounced dead two and a half hours later, at 8:50 p.m. He was 32 years old.
Abe's son Maiki followed his father into motorcycle racing, competing in the All Japan Road Race Championship. Maiki attended the Yamaha VR46 Mastercamp training at Valentino Rossi's ranch in Tavullia, Italy, accompanied by his grandfather Mitsuo. In 2023 Maiki joined VFT Racing Yamaha to compete in the Supersport World Championship, extending the Abe family's connection to professional motorcycle racing into a third generation.
Norick Abe is remembered as one of the most naturally gifted and expressive riders of the 500cc era, whose brief but brilliant career inspired riders well beyond his native Japan.