Varhaug began motorsport in 2005, finishing fifth in the Norwegian Championship ICA Junior category and fourth in the Viking Trophy ICA Junior class. In 2006, he went a step further by winning the Norwegian Championship ICA Junior title, establishing himself as a promising junior talent.
Varhaug stepped into single-seaters in 2007 with Jenzer Motorsport in the Swiss Formula Renault 2.0 championship, taking four race wins and six additional podiums to finish as runner-up. He also appeared in selected rounds of the Italian Formula Renault 2.0 and Formula Renault 2.0 NEC championships that year.
He continued with Jenzer in 2008, competing in full seasons of both the Eurocup Formula Renault 2.0 and Italian Formula Renault 2.0. In the Eurocup he finished fifteenth overall, but in the Italian series he was far more dominant, scoring points in all fourteen races and taking three victories to claim the title comfortably. At the end of the season, Varhaug tested Formula Renault 3.5 machinery at Paul Ricard with Prema Powerteam, completing over one hundred laps.
In 2009, Varhaug moved to the International Formula Master series with Jenzer Motorsport, finishing fifth in the standings. He was runner-up in the rookie classification, finishing behind Alexander Rossi. He achieved five podium finishes and a pole position at Brno during the campaign.
Varhaug graduated to the newly formed GP3 Series with Jenzer Motorsport in 2010, the category's inaugural season. In the opening race of the championship at Barcelona, he qualified on the front row and then led from the start to win, becoming the first-ever GP3 race winner, finishing ahead of Robert Wickens and Esteban Gutiérrez. The result was a landmark moment in the short history of the series.
The rest of the season proved harder, and Varhaug did not score another championship point, ending the year thirteenth overall. He was later demoted from what appeared to be a third-place finish at Spa-Francorchamps after overtaking the safety car. He also posted the fastest lap in the final race at Monza but could not convert it into a top-ten finish and therefore received no bonus point.
Varhaug returned to the GP3 Series in 2014 with Jenzer, finishing seventeenth in the standings.
Varhaug stepped up to the GP2 Series for 2011, joining DAMS alongside Romain Grosjean. In the GP2 Asia Series at the start of the year, he finished thirteenth while Grosjean won the Asia title, and DAMS took the teams' championship. In the main 2011 GP2 Series season, Varhaug failed to score points and was classified twenty-third overall, while Grosjean claimed the drivers' title. After losing his DAMS seat, Varhaug moved to the Auto GP World Series with Virtuosi UK, ending his time in the senior feeder categories.