Dominant Shiffrin moves to within one win away from Vonn's record
Shiffrin was in control from the off, laying down the fastest opening run in 48.93 seconds.
The 27-year-old American followed that up with a fourth-fastest second leg, in 47.49 seconds, for an aggregate of one min 31.42 seconds, finishing 0.76 seconds ahead of Slovak rival and Olympic champion Petra Vlhova.
Sweden's Anna Swenn-Larsson rounded out the podium, at 1.21 seconds.
"I had so much fun skiing today and it was really my best skiing - both runs," Shiffrin said.
"Nothing less than the best is going to work and I was taking all the risks I needed and then nailed it on the finish."
The win left Shiffrin atop the slalom standings on 525 points, with Switzerland's Wendy Holdener - fourth in Zagreb - in second on 420 points ahead of Vlhova (360) and Swenn-Larsson (320).
Shiffrin unsurprisingly also tops the overall standings, with 975 points, Vlhova sitting a distant second on 586.
The victory was Shiffrin's 81st on the World Cup circuit and her fifth consecutive triumph in a purple run of form ahead of next month's World Ski Championships in the French resorts of Courchevel and Meribel.
Shiffrin's winning streak includes victories in a super-G in St Moritz, two giant slaloms and a slalom in Semmering, and now the first of the two Zagreb slaloms in unseasonably warm weather.
The conditions resulted in the lowering of the start gate and soft snow underfoot, which caught out many of the top athletes in the first run on the steep pitch towards the end of the course.
"I'm really happy with how my skis felt in these conditions," Shiffrin said.
"I think that was the most exciting thing of the day, that I felt so good skiing when it's a bit softer."
Apart from having Vonn's record in her sights, Shiffrin, who now has 51 slalom wins, is also closing in on the overall record of 86 World Cup victories set by Ingemar Stenmark, who retired back in 1989.
The American will go again on Thursday in the second of the Zagreb slaloms with a massive opportunity to draw level with Vonn.
