Kyle Palmieri and Mathieu Perreault tallied a goal apiece during the early Anaheim flurry, which staked the West's top seed to a 4-0 lead just over nine minutes into the middle stanza.
Jamie Benn and Colton Sceviour scored 1:33 apart later in the second to slice the Stars' deficit in half before Tyler Seguin's tip-in with 6:07 left in the third brought the visitors within one.
However, rookie Frederik Andersen, who got the start in net for Anaheim over Jonas Hiller, turned away all comers over the final six minutes to help the Ducks grab a 1-0 lead in the best-of-seven series.
Andersen finished with 32 saves in his postseason debut for the Ducks, who used a 54-20-8 regular-season record -- the best in franchise history -- to secure the top spot in the West.
Sceviour also added an assist in the setback, while Kari Lehtonen allowed four goals on 35 shots for the Stars, who claimed the West's second wild card spot to reach the playoffs for the first time since 2008.