Welti began racing in the Swiss Sports Prototype Championship, which he won in 1980 driving a Wittwer March BMW. After stepping back from active competition, he joined Sauber as its first team manager, a role that would shape the team's trajectory through the golden age of Group C racing.
Under Welti's management, Sauber forged a partnership with Mercedes-Benz that became one of the most dominant forces in sports car racing. In 1986, Mike Thackwell and Henri Pescarolo delivered the team its first major victory at the Nürburgring 1000 km. With AEG backing arriving in 1988, Sauber-Mercedes battled Jaguar for the World Sportscar Championship and the Silver Arrows livery returned to competition in 1989 — the first time since 1956. Jean-Louis Schlesser won the Drivers' World title that season as the team claimed seven of eight races in the Constructors' standings. The 1989 24 Hours of Le Mans brought a 1–2 finish, with Jochen Mass, Manuel Reuter, and Stanley Dickens taking victory ahead of Mauro Baldi, Kenny Acheson, and Gianfranco Brancatelli. In 1990, Schlesser and Baldi shared the Drivers' title while Jochen Mass alternated alongside Mercedes Juniors Michael Schumacher, Karl Wendlinger, and Heinz-Harald Frentzen; the team secured its second consecutive Constructors' crown.
As sports car racing declined in 1991, Welti moved to Porsche Motorsport as F1 Project Manager, overseeing a brief and troubled partnership with the Footwork Formula One team. The Porsche engine program proved uncompetitive and Welti withdrew the project. He subsequently reorganized Porsche's global motorsport department and in 1993 launched the Porsche Supercup as a Formula One support series, personally negotiating its placement within the F1 calendar with Bernie Ecclestone.
In 1994, Welti directed Porsche's return to Le Mans with a modified 962, working alongside racing engineer Norbert Singer. The program developed, built, and tested two race cars and a reserve car in record time. Mauro Baldi, Hurley Haywood, and Yannick Dalmas brought the car to victory, giving Welti his second 24 Hours of Le Mans win — this time as race director rather than team manager.
In May 1995, Welti returned to Sauber as director of its Formula One operation. He played a central role in restructuring the team's organization, securing the Petronas title sponsorship from Malaysia, and establishing the Ferrari engine supply relationship that began in 1997. During his tenure the team achieved its first Formula One podium finish.
From 2000, Welti led the establishment of V8STAR, a European silhouette touring car series featuring manufacturer bodywork from Jaguar, Volkswagen, Opel, Ford, Lexus, Audi, and BMW over uniform V8 engines. The series ran in Germany and attracted established racing talent.
In 2004, Welti became franchise holder and Team Principal of A1 Team Switzerland in the A1 Grand Prix World Cup of Motorsport, a national-teams championship structured to provide drivers from smaller motorsport nations with high-level single-seater competition. With Neel Jani as lead driver, A1 Team Switzerland won the World Cup in the 2007–2008 season and finished as runner-up in 2006 and 2009. The team also gave early international opportunities to Marcel Fässler, Sébastian Buemi, Alexandre Imperatori, Tom Dillmann, Rahel Frey, and Natacha Gachnang. No other team nor driver matched Switzerland and Jani's overall record in the championship.
In 2013, Welti joined the Volkswagen Group Motorsport Steering Committee and was deployed to Italy as Motorsport Director of Lamborghini Squadra Corse, tasked with building a new motorsport department from the ground up. After completing that assignment he served as a traveling representative for the group across European and American racing events, conducting negotiations with organizers, promoters, federations, and technical partners across multiple series.
Between 2018 and 2020, Welti served as a strategic motorsport advisor to BMW, attending major events in the WEC, IMSA, Formula E, DTM, and WRX, including endurance classics at the Nürburgring, Spa, Le Mans, Sebring, and Daytona. Since 2019, he has also served as Sporting Director of the SuperCharge all-electric racing series.
In 1990, Welti received the BP Racing Trophy, an award for special services to motorsport recognizing his results with Sauber-Mercedes — a prize first awarded in 1965. Following A1 Team Switzerland's 2007–2008 world championship, the team was awarded the Auto Sport Award for services to Swiss motorsport in December 2008, joining previous recipients including Jo Siffert, Clay Regazzoni, and Peter Sauber. In 2010, Welti was inducted into the Wall of Fame at the Autobau museum in Romanshorn as a Swiss legend of motor racing.