Bjorkstrand and Gourde, who had the primary assist on the go-ahead goal, were acquired by the Lightning from the Seattle Kraken as part of a three-team trade also involving the Detroit Red Wings on Wednesday.
"We had a few chances, and I want to be around the paint and help create some stuff," Gourde said. "I was so excited for Oliver to get that goal."
Said Lightning coach Jon Cooper: "I thought both guys...they played three time zones away just 48 hours ago and you can make an argument they were two of our best forwards out there."
Rasmus Dahlin and JJ Peterka each had a goal and an assist for the Sabres (24-31-6), who have lost five straight (0-4-1). Ukko-Pekka Luukkonen made 20 saves.
"I didn't [like] the first five minutes of the game, but after that I thought we skated well and made a lot of plays," Buffalo coach Lindy Ruff said. "It was enough that we should've gotten a win."
Nick Paul gave Tampa Bay a 1-0 lead at 5:01 of the first period when his centering pass attempt deflected off Connor Clifton's skate and in.
Dahlin tied it 1-1 at 5:17 when he took control of the puck in the neutral zone and snapped a wrist shot past Vasilevskiy glove side.
"Too many selfish plays, I had a stupid penalty, it's bad," Dahlin said. "We had a ton of chances, we just couldn't bear down when we got the chances. We let them come into our end and score easy goals. It's frustrating."
Darren Raddysh put the Lightning ahead 2-1 at 6:53 with a shot from the point that deflected off Bowen Byram's stick and over Luukkonen.
Alex Tuch tied it 2-2 at 8:20 on a shot from the right circle that beat Vasilevskiy to the far post. It was his fifth goal in the past six games.
Peyton Krebs gave Buffalo a 3-2 lead at 17:09 with a shot from the hashmarks off a pass from Ryan McLeod.
Guentzel tied it 3-3 at 58 seconds of the second period when he chipped the puck past Luukkonen from the doorstep off a pass from Erik Cernak.
"We stuck with it,” Guentzel said. “We know they are a high-powered offense, so you just have to stay with them. One of those games that just happens throughout the year."