Mast grew up in racing as his father and uncle were both race team owners. He began racing at age 16 at Natural Bridge Speedway and Eastside Speedway, after he traded an Angus cattle for his first car. After racing at the local track level for the decade, Mast began running the Busch Series in 1982, and had four top-ten finishes in 11 eleven starts in his No. 22. Mast's first full-time season came in 1985, where he had fifteen top-ten finishes and finished seventh in the season points. Two years later, he won his first NASCAR race, at the Grand National 200, then followed it up with another win the next week. He finished 11th in points that year. He improved to eighth position in 1988 the same year he made his Winston Cup debut for Buddy Baker at the Busch 500, finishing 28th at that race. Mast won five Busch races while running full-time the next two years, before focusing his efforts on the Cup Series.

Mast ran 13 races for Mach 1 Racing in 1989, finishing sixth at the Daytona 500 in an unsponsored car, which Mast called his proudest achievement in racing. He still believes he would have won had his team been willing to gamble on fuel mileage. Mast ran selected races in 1990 for D.K. Ulrich before finishing the year with Travis Carter Motorsports. In 1991, Mast signed to drive the No. 1 Skoal Classic Oldsmobile for Richard Jackson's Precision Products Racing. He started out the season by leading 14 laps in the Daytona 500 and finished fourth. He had three top-tens and finished 21st in points. That year, the Talladega Superspeedway produced a couple of highlights for Mast. In the Winston 500, he pushed a fuel-deficient Harry Gant (driving for Leo Jackson, Richard's brother) during the final lap of the race, helping Gant win (Mast was one lap down in 10th). This action is prohibited after the white flag by NASCAR rules, regardless of who the individual drivers are, but he was not fined money or points. With less than 25 laps to go in the DieHard 500, Mast was tapped by Buddy Baker entering the tri-oval and flipped over. He slid to a stop a few hundred feet beyond the start-finish line and soon climbed out of the car, much to the delight of the crowd. He was not injured, but half-jokingly said afterwards, "I'm okay but I need another pair of underwear". The next year, Mast won his first career Cup pole at the final race of the 1992 season, the 1992 Hooters 500, which was Richard Petty's final race, Jeff Gordon's first race, and the day that Alan Kulwicki won the championship by one race position over Bill Elliott. Mast's race ended on the first lap in a crash. The team switched to Ford in 1993. Mast had a career year in 1994, with ten top-ten finishes and a career-high-tying eighteenth, finishing a career-best second at Rockingham Speedway, a race where he slid sideways while racing side-by-side with winner Dale Earnhardt coming out of the final corner. In August of that season, he won the pole position at the inaugural Brickyard 400 at the Indianapolis Motor Speedway (a race for which 90 cars were entered), finishing eighteenth in points. In comparison, 1995 was disappointing for Mast, with only three top-tens. Skoal left at the end of the season, and Hooters replaced them, as the team switched to Pontiac. He had three top-tens late in the year, but when the season came to a close, he and Hooters left PPR.

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