Team Sky rider Boasson Hagen finished ahead of IAM rider Sebastien Hinault in a frantic end to the 190km stage from Fredrikstad to Sarpsborg.
"The sprint with Boasson Hagen was head to head until the very last metres, where I was able to pass him," said Kristoff.
"I'm very happy with this victory: it is the first time I have taken part in the Tour of Norway and it's a pleasure to wear the leader's jersey at my first attempt thanks to this brilliant performance. I want to keep it as long as possible.
"Tomorrow I think there will be another sprinters' stage, suited to my characteristics. I'm in good shape so I'll try for another win, for sure."
Thursday's second stage is a 179km run from Kongsberg to Skien. The five-stage race finishes on Sunday.
1 KRISTOFF Alexander Katusha Team
2 BOASSON HAGEN Edvald Sky Procycling
3 HINAULT Sébastien IAM Cycling
4 BOS Theo Blanco Pro Cycling Team
5 SBARAGLI Kristian Team MTN Qhubeka
6 VAN ASBROECK Tom Topsport Vlaanderen – Baloise
7 DRUCKER Jean-Pierre Accent.jobs – Wanty
8 VANDOUSSELAERE Sven Topsport Vlaanderen – Baloise
9 LASCA Francesco Caja Rural-Seguros RGA
10 SISKEVICIUS Evaldas Sojasun