Mikael Backlund scored to reach 600 career points and Dustin Wolf made 27 saves for the Flames (26-32-7), who had lost six of their past seven games (1-5-1).
Jack Hughes and Luke Hughes each had a goal and an assist, and Jesper Bratt had two assists for the Devils (32-31-2), who have lost two straight after four wins a row and are 3-2-0 five games into a season-long, seven game homestand. Jacob Markstrom made 24 saves in his seventh straight start.
Beecher gave the Flames a 1-0 lead at 5:04 of the first period with a wrist shot from the right face-off circle, getting the pass from Matvei Gridin to break out on a 2-on-1 with Matt Coronato.
Kevin Bahl pushed it to 2-0 on a slap shot from the left point that deflected off the stick of Devils forward Paul Cotter in the left circle at 14:07.
Maxim Tsyplakov scored his first goal with the Devils in his 11th game on a snap shot from the right hashmarks to pull New Jersey within 2-1 at 16:14. Tsyplakov was acquired in a trade with the New York Islanders in exchange for Ondrej Palat on Jan. 27.
Beecher scored his second of the game, a short-handed goal, to give Calgary a 3-1 lead 54 seconds into the second. He stole the puck in the neutral zone, skated down the right side, and banked a wrist shot off Markstrom before putting his own rebound in between Markstrom's left skate and the right post.
Just 42 seconds later, Simon Nemec pulled the Devils within 3-2 at 1:36 with a power-play goal on a rebound after Luke Hughes split two defenders and took the initial shot from the slot.
Yegor Sharangovich scored on the backhand from the slot off a 2-on-1 with Ryan Strome at 16:48 to give the Flames a 4-2 lead.
Backlund made it 5-2 on a wrist shot from the slot at 4:10 of the third period. Backlund has 600 points (229 goals, 371 assists) in 1,131 games, all with the Flames.
Jack Hughes made it 5-3 at 9:37. Hughes hopped on the ice with fresh legs and snapped a shot past Wolf far side.
Luke Hughes scored a power-play goal on a wrist shot from the high slot with Markstrom pulled for a 6-on-4 advantage for the 5-4 final at 19:56.
