No goals were scored in the first period and the clubs traded power-play opportunities in the second.
Buffalo was afforded the first chance on the power play 3:17 into the second and while they generated only one shot on goal, both Jack Eichel and Reinhart hit the post as the Sabres maintained possession in the Devils end for the full two minutes.
The Devils went on the man advantage at the 5:45 mark of the middle frame and Riley Sheahan in particular did a good job to kill it off.
Buffalo then took the lead when they went back on the power play minutes later. Eichel had hit the post earlier in the advantage, but Olofsson was able to finish the job, potting one with 7 seconds left in the power play and 9:40 remaining in the period.
At the 5:16 mark of the third period, Asplund tipped a high Reinhart shot out of the air and into the back of the net to make it 2-0.
Hall fed Cozens from behind the net just 1:34 later to put Buffalo ahead by three.
The Devils had one final opportunity on the power play with 2:50 left in regulation. Ullmark made a sprawling desperate glove save with 1:05 remaining on P.K. Subban to preserve the shutout, but after a sequence of heavy pressure (and with the extra attacker), the Devils finally put one in with 27.9 seconds left.
Eakin's goal - his first as a Sabre - was an empty netter just before time expired to make it 4-1.