Kristoffersen timed 55.87 seconds on the Ganslern course to lead Michael Matt of Austria by 0.10, while Hirscher was 1.05 behind in third. Fourth-place Daniel Yule of Switzerland had 1.29 to make up in the final run.
"It's only one tenth ahead of Michi," Kristoffersen said. "It was OK, a nice run. It's turning a bit, pretty difficult, but it's not a fast course. You have to push."
Kristoffersen set a World Cup record this season by getting podiums in all six slaloms without winning a race. He has been after his 15th career slalom victory since triumphing in Schladming, Austria, a year ago.
Continuing their slalom rivalry from past seasons into next month's Pyeongchang Olympics, Kristoffersen was runner-up to Hirscher in five races this season, including three giant slaloms.
On Sunday, many racers struggled in the tough conditions after overnight snowfall. On a difficult course set by his coach Jani Hladnik, Russia's Alexander Khoroshilov straddled a gate and failed in his bid to improve his slalom ranking ahead of the Olympics.
Britain's Dave Ryding, who finished runner-up to Hirscher last year, trailed Kristoffersen by 2.89 seconds in 25th.
With a win on Sunday, Hirscher would match Hermann Maier's Austrian record of 54 World Cup victories. Only Swedish standout Ingemar Stenmark (86) won more races.
A six straight win would leave the Austrian still three victories short of the best mark of nine consecutive slalom wins, set by Italian great Alberto Tomba in 1995.
Leading the standings by 174 points over Kristoffersen, Hirscher was on course for an unprecedented seventh overall title.