The Katusha sprinter arrived at the finish of stage 12 in Saint-Étienne as part of a reduced bunch of 60 riders, whittled down over a hilly 185.5 kilometres between Bourg-en-Bresse and the capital of the Loire département.
Kristoff's biggest win to date had come at the 2014 Milan San-Remo in March, where he beat Trek's Fabian Cancellara and Team Sky's Ben Swift to take the Italian one-day Classic.
In Saint Etienne on Thursday, it was Peter Sagan and Arnaud Démare who were left disappointed thanks to Kristoff's fast finish, while his many fans back home in Norway were left elated.
"I hope they have a big party," Kristoff grinned, "and that the whole country celebrates. I know how big the Tour is for people in Norway. This is the race that everyone knows, and I've always dreamed about winning a Tour stage, so it feels great to have finally won here."
Kristoff revealed that he'd specifically targeted the Saint Etienne stage, and used an old Tour trick to ensure that he came to the start in Bourg-en-Bresse feeling as refreshed as possible in the hope of achieving his goal.
"Yesterday [Wednesday's 11th stage] I took it easy with a view to being good today, so maybe that was the key to my success. I'd tried to save my legs on the hills, which meant that I felt pretty good on the climbs today, and was never really on the limit," Kristoff explained. "But coming towards the finish, anything can happen – you can easily get boxed in – so I was a little bit nervous.
"A lot of teams wanted to be at the front, so we were a little squeezed, and I lost Luca [Paolini] and then [Alexander] Porsev, too, so I just had to find my way to the front by myself," said Kristoff. "Quick Step were giving [Matteo] Trentin a good lead-out, so I got on his wheel. I felt then like I could start my sprint whenever I wanted to, and I think I got it just about right..."