Asch began his motorsport career in 1973 in slalom competition before transitioning to hillclimbing, where he won the German Hillclimbing Championship in 1981. After racing in the lower Trophy division of the Deutsche Rennsport Meisterschaft, he made his DTM debut at the age of 34 in 1985 — a relatively late start for a touring car career that would nonetheless span the better part of twenty years.
From 1985 to 1994, Asch raced for Mercedes in the DTM, establishing himself as one of the series' recognizable regulars even if championship wins eluded him. Alongside his touring car work, he built a parallel career in Porsche competition, winning the German Porsche 944 Turbo Cup in both 1988 and 1989, then adding the German Porsche Carrera Cup title in 1991.
When the original DTM folded after 1994, Asch moved to the German Supertouring Championship (Super Tourenwagen Cup, or STW), where he raced from 1995 through 1999. In his fifties, he returned to Porsche competition, contesting the German Porsche Carrera Cup through 2003 with a single additional appearance in 2004.
Also in 2004, Asch raced in the European Touring Car Championship for RS-Line Ford — the same brand for which he operates a dealership in Ammerbuch. From 1999 to 2005 he served as lead driver for the Japanese Falken team's Nissan Skyline GT-R entry at the 24 Hours Nürburgring, making a final appearance at that race in 2010 as part of a Sport Auto magazine project.
Asch is as well remembered for two controversial contact incidents as for his race wins.
The first came in 1994 at the Alemannenring during the DTM season finale. Alfa Romeo driver Alessandro Nannini, running ahead of the already-lapped Asch, momentarily locked his rear wheels entering a hairpin before regaining the racing line. Asch then steered deliberately into Nannini's car, spinning him out of contention. As a direct consequence, Asch's Mercedes teammate Klaus Ludwig claimed the championship. After pitting, Nannini rejoined, caught Asch, and rammed him off the road at the same hairpin in retaliation.
The second incident closed the final STW season at the Nürburgring in 1999. Championship contenders Uwe Alzen and Christian Abt were leading when Alzen's Opel was damaged in contact with Abt's Audi teammate Kris Nissen. Asch, who had been called in for a jump-start penalty — which he claimed not to have received because his radio had failed — then struck Abt's car in the final corner. The collision promoted Alzen to second, handing him the STW title. Though initially ruled a racing accident, a DMSB court of appeal later reversed the decision based on video evidence and awarded the championship to Abt. Alzen's celebration was rescinded in November, and Asch was sanctioned for the deliberate move.
In his later years, Asch directed his racing energy toward supporting his son Sebastian Asch, who went on to compete in the SEAT León Cup and later in German touring car categories.