The Kings have lost five in a row while St. Louis captured its fourth straight.
Elliott made 30 saves in what was his 24th career shutout. This is his seventh NHL season and his third with the Blues.
Brenden Morrow scored the only first-period goal on the power play and Oshie then broke things open by scoring twice in a span of less than two minutes.
Vladimir Tarasenko and Barret Jackman added third-period goals. It was the first goal of the season for Jackman and only the 24th in a career that has stretched more than a dozen campaigns and 683 games.
Martin Jones opened in goal for the Kings, but was removed after giving up two goals on 14 shots.