The Rams (3-2) scored 23 points in the final 8:36 of the game after the Seahawks (3-1) held a 27-10 lead with just over 5 1/2 minutes left in the fourth quarter.
Bulger was 24-of-42 for 325 yards with three touchdowns and three interceptions.
The Rams got within 10 points with 5:34 left, as Bulger threw a pass over the middle that Brandon Manumaleuna caught in the end zone for an eight-yard TD reception. That made the score 27-17, and marked the first time this season that the Seahawks have allowed more than one touchdown in a game.
Next, Bulger hit Kevin Curtis for a touchdown, and the Rams were within 27-24 with 3 1/2 minutes left.