Tissot, seeking his first World Cup victory, covered the relatively flat layout in 50.43 seconds during a steady snowfall.
Finland's Kalle Palander, who won Sunday's giant slalom on the Gran Risa, was second, 0.12 seconds behind. Defending overall World Cup champion Benjamin Raich of Austria was third, 0.18 back.
Olympic combined champion Ted Ligety of the United States was fourth, 0.25 out. Ligety failed to finish the opening two slaloms of the season.
Michael Janyk of Canada, who leads the slalom standings, was eighth.
Tissot's coach set the course for the opening leg. The Frenchman's best career results were second in slaloms in Beaver Creek, Colorado, and Are, Sweden, last season. He finished fourth in this season's opener in Levi, Finland, last month.
Miller hasn't won a slalom since Dec. 13, 2004, in Sestriere. The last time he finished a slalom was in January in Kitzbuehel, Austria, placing 18th.
Because of his struggles, Miller started outside the top group. He looked out of control right from the start and skied off course after about 10 gates. He gathered himself and started again, only to fall a few gates later, reports AP.
Miller still had something to celebrate. Courtesy of his super-G victory in Val Gardena on Friday and second-place finish in Sunday's giant slalom, Miller had already clinched the Ђ25,000 (US$33,000) Dolomiti Super Trophy awarded to the weekend's overall winner.
Miller said he may give the money to charity.
Indeed, how dare they run US-independent policy? They should have followed the example of the European Union that turned independent states of the Old World into US-ditto entities