Los Angeles was in position to pick up what would have been a vital two points but Jonathan Quick gave up two quick goals and finished the contest with only 11 saves in 15 shots.
Filppula tied it at 3-3 by tipping in a shot and Helm scored the go-ahead goal on a tip-in with 1:13 to play in regulation. Henrik Zetterberg scored the other two Detroit goals, one in each of the first two periods.
Dwight King had put Los Angeles in front at 3-2 less than 2 minutes before Filppula produced his goal.
The Kings missed a chance to move in front of San Jose for the eighth and final playoff spot in the Western Conference.
Joey MacDonald got the win by making 21 saves.