Alsup made his CART debut at the Arizona Republic/Jimmy Bryan 150 at Phoenix International Raceway, driving a McLaren-Offenhauser for WASP Racing and finishing eleventh from last on the starting grid. His attempt to qualify for the 1979 Indianapolis 500 did not result in a race start. He attempted three cars during qualifying — a McLaren-Offenhauser for WASP Racing, a Sta-On Eagle-Offenhauser for Gehlhausen Racing, and a Penske PC7-Cosworth DFX for Team Penske — but USAC officials disqualified the Penske entry after ruling that the engine used had come from teammate Bobby Unser's car.
He completed the 1979 CART season with WASP Racing, accumulating consistent if unspectacular results that were sufficient to earn him the first CART Rookie of the Year Award.
In 1980, Alsup acquired the inventory of WASP Racing and established Alsup Racing. Running a Penske PC7-Cosworth DFX under his own banner, he opened the season with a sixth-place finish at Ontario Motor Speedway, posted a third-place result at the Red Roof Inns 150 at Mid-Ohio Sports Car Course, and finished seventh in the overall CART standings. His progress that year was recognized with the CART Most Improved Driver Award.
The 1981 season represented the high point of Alsup's career. He drove for Team Penske alongside Bobby Unser and Rick Mears, competing in a Norton-Spirit sponsored car as well as his own entry. He recorded three third-place finishes: at the Norton Michigan 500 at Michigan International Speedway (where he also led three laps), at the Los Angeles Times 500 at Riverside International Raceway, and at the AB Dick Tony Bettenhausen 200 at Wisconsin State Fairgrounds Park Speedway. Those results, combined with consistent points-scoring, elevated him to second in the championship standings — a career-best that was not enough to retain his Penske seat for the following season.
He also competed in the 1981 Indianapolis 500, where he finished eleventh.
Returning to his own team in 1982, Alsup drove a Penske PC9B-Cosworth DFX and achieved a best finish of fourth at the Stroh's 200 at Atlanta Motor Speedway, dropping to eleventh in the championship. In 1983 and 1984 he worked with Argo Racing Cars and drove March chassis, failing to qualify for the Indianapolis 500 in both years and scoring no championship points. His final race start came at the Molson Indy Montreal at Sanair Super Speedway in 1984.
Alsup died on August 9, 2016, in a crane accident near Silverton, Colorado, where he had been a long-term resident.
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