In the final game for both clubs before the Olympic break, the Kings snapped a four-game losing streak with their second victory in 12 games.
Bobrovsky, last season's Vezina Trophy winner, stopped 18 shots and lost for just the third time in 13 starts. Jack Johnson scored Columbus' goal during a power play early in the first period.
Regehr used teammate Jeff Carter and Blue Jackets defenceman Denis Savard as a screen before slapping his second goal of the season past Bobrovsky from the top of the left circle.
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