Rookie Wyatt Johnston also scored a goal and Jamie Benn added two assists for the Stars.
Eichel and Michael Amadio both scored goals for Vegas, which saw its five-game home winning streak snapped.
Patrik Laine had a goal with two assists and Kirill Marchenko scored twice for the Blue Jackets, who swept the two-game season series from Edmonton.
It was Oshie's first career game with a goal, assist and a fight, known as a Gordie Howe hat trick.
Tyler Toffoli scored, Jacob Markstrom turned away 18 shots and Nazem Kadri had an assist in his return to Colorado.