Rudd grew up in southeastern Virginia, the son of Alvin R. Rudd Sr., the president of Al Rudd Auto Parts. He began racing as a teenager in karting and motocross before attempting stock car racing at age eighteen, making his NASCAR debut at Rockingham in 1975 driving the No. 10 Ford for family friend Bill Champion. Qualifying 26th, he finished eleventh despite running 46 laps down. He ran three additional races that season, his best finish a tenth place at Bristol.
Rudd ran four more races in 1976 for his father, posting a tenth-place finish at the Firecracker 400. He went full-time in 1977 in the No. 22 car, again owned by his father, collecting ten top-ten finishes and winning the series Rookie of the Year award. A part-time 1978 campaign still yielded four top-ten finishes. In 1979, Rudd signed with Junie Donlavey to drive the No. 90 Truxmore car, earning four top-five finishes and ninth place in the final championship standings.
In 1981, Rudd joined DiGard Motorsports in the No. 88 car. He had no victories but won his first three career pole positions and began a consecutive race starts streak that would eventually reach 788, a record at the time of his retirement. In 1982, he moved to Richard Childress Racing in the No. 3 Pontiac, recording six top-five finishes and finishing ninth in points.
The 1983 season marked Rudd's emergence as a confirmed race winner. At the Daytona 500 that year, he became the youngest pole winner in Daytona 500 history at age 26, a record that stood until 2014. He won twice during 1983 — at Riverside and Martinsville — and also won his debut Busch Series race at Dover Downs that season.
In 1984, Rudd and Dale Earnhardt swapped rides. Rudd moved to the No. 15 Ford operated by Bud Moore, while Earnhardt took the Childress seat. The season opened with a severe crash in the Busch Clash at Daytona, in which Rudd suffered a concussion and torn cartilage in his rib cage. With his eyes swollen shut, he taped his eyelids open to compete in the Daytona 500 the following week, also wearing a flak jacket to protect his injured ribs. He won for Bud Moore in only his second start for the team at Richmond. The incident prompted NASCAR to institute medical examination policies for drivers involved in race-weekend crashes.
Rudd remained with Bud Moore through 1987, improving to seventh in points in 1984 and a career-best fifth in 1986. He won twice in 1987 before departing the team at season's end.
The period from 1983 through 1987 established the foundation of Rudd's record sixteen-season consecutive winning streak. Beginning with his victories at Riverside and Martinsville in 1983, Rudd won at least once each year through 1998. The streak, achieved across multiple team affiliations and including a stint as an owner-driver, reflected both his technical precision and an ability to extract results from varied equipment. During the Bud Moore years, Rudd ranked among the most reliable performers in the series without yet carrying the statistical weight that later periods of his career would add.
Rudd was inducted into the Virginia Sports Hall of Fame in 2007 and the Hampton Roads Sports Hall of Fame in 2010. He was inducted into the NASCAR Hall of Fame in 2025. His record of 788 consecutive starts stood as NASCAR's "ironman" mark until Jeff Gordon surpassed it in 2015.