Bratt scored on a wrist shot from the slot to extend the lead to 3-0 at 18:51.
Olivier pulled Columbus to within 3-1 on a deflection from the slot at 3:30 of the second period.
"Our line (with Olivier and Kent Johnson) has been going pretty good," center Boone Jenner said. "We want to play to our strengths, get it down there and kind of grind them below the goal line, protect pucks and bring it to the goal line that way."
Kirill Marchenko cut it to 3-2 when he scored with his backhand while skating through the slot at 7:16.
Timo Meier extended the Devils' lead to 4-2 at 16:49 on a wrist shot from the right face-off circle.
"We were a little bit too much on our heels in the second, but all four lines were going; everybody chipped in," Meier said. "That's a playoff mentality. That's how we got to approach games. This definitely feels good, but now we got to build on this one."
Luke Hughes scored on a wrist shot from the top of the left face-off circle 26 seconds into the third period to push the lead to 5-2.
Olivier scored on a wrist shot at the left post that deflected off the stick of Bratt at 10:29 for the 5-3 final.
"We're in a playoff race here and we had an opportunity to do some damage in the standings and we didn't," Olivier said. "We'll analyze, shake it off, and get back to it in the next game (against the Vegas Golden Knights on Thursday)."
NOTES: Olivier has five goals and two assists in his past seven games. ... Hughes (16 goals, 64 assists in 141 games) is the fourth defenseman age 21 or younger in Devils/Colorado Rockies/Kansas City Scouts history with at least 80 career points. He joins Scott Niedermayer (106 points), Barry Beck (102) and Joe Cirella (88).