Jani began karting in 1998 before moving to Formula Renault 2000 Eurocup in 2001 and the Formula Renault V6 Eurocup in subsequent seasons. In the 2003 Formula Renault V6 Eurocup he finished second by only four points driving for Jenzer Motorsport. He graduated to the GP2 Series in 2005 with Racing Engineering, winning races at the Hungaroring and Monza. In 2006 he substituted for the injured Nicolas Lapierre at Silverstone and Magny-Cours for Arden, becoming the only driver recorded to have competed in both GP2 and Formula One on the same day.
Jani represented A1 Team Switzerland throughout most of the A1 Grand Prix series' existence. In the inaugural 2005-06 season the team earned second place overall. He drove all the races for Switzerland in 2007-08, taking four wins and 168 points to claim the series championship. The team finished runner-up again in 2008-09.
Jani served as Scuderia Toro Rosso's third driver for the 2006 Formula One season alongside race drivers Scott Speed and Vitantonio Liuzzi. He was subsequently linked to reserve roles at Red Bull Racing and Force India but did not secure a race seat. In June 2023 he signed with Audi as a simulator driver to help develop their power unit in preparation for their 2026 Formula One entry.
For 2007, Jani drove for PKV Racing in the Champ Car World Series, ending the season ninth overall with 231 points. The series was absorbed by the IRL before the following season.
Jani joined Rebellion Racing for sports car competition from 2010, initially in the FIA Le Mans Series partnered with Nicolas Prost, and moved to full-time WEC racing in 2012 alongside Prost and Nick Heidfeld. After joining the Porsche works LMP1 programme in 2014, he took his first LMP1 victory at Sao Paulo and steadily accumulated podiums in subsequent seasons.
The defining moment of his career came at the 2016 24 Hours of Le Mans. Jani and co-drivers Romain Dumas and Marc Lieb inherited victory in dramatic circumstances when the leading Toyota of Kazuki Nakajima broke down on the final lap with the finish in sight. He had opened the 2016 WEC season with a win at Silverstone and a second at Spa before the Le Mans result.
Jani made his Formula E debut in the 2017-18 season with Faraday Future Dragon Racing alongside Jerome d'Ambrosio, but quit after two 18th-place finishes at the Hong Kong season opener to focus on WEC commitments. He returned to the championship with Porsche for their maiden Formula E season in 2019-20, racing alongside Andre Lotterer through to the end of the 2019-20 season before being replaced by Pascal Wehrlein.