Cozens received a backhand pass from JJ Peterka low in the left circle and scored glove side to cut it to 2-1 at 13:52 of the second period.
Johansson had to be sharp early in the third period on a Tuch breakaway in the first minute, making a pad save.
“All game he was there making huge stops for us when we needed them,” Lightning center Anthony Cirelli said. “Obviously that one in the third there to start the period, the breakaway there, was massive for us. Whenever he’s in there he makes huge stops for us and gives us a chance to win every night.”
Calvin de Haan scored an empty-net goal at 18:44 of the third period for the 3-1 final.
Buffalo outshot Tampa Bay 15-2 in the third.
“We knew we just had to get it behind them and work,” Cozens said. “Make them turn it over and spread the zone and just get pucks to the net and try to have guys around the net. We didn’t do a good enough job at that.”
NOTES: The Lightning have scored 49 goals in the first period this season, third-most in the NHL behind the Vancouver Canucks (57) and Edmonton Oilers (51). … Hagel has 11 points (five goals, six assists) in seven career games against the Sabres, who selected him in the sixth round (No. 159) of the 2016 NHL Draft. … Luukkonen’s shutout streak surpassed the run of 158:34 by Joey Daccord of the Seattle Kraken from Dec. 29-Jan. 4. … Rasmus Dahlin, who played in his 400th NHL game, had an assist to bring his career total to 270 points (59 goals, 211 assists). Only four other active defensemen have posted as many points through 400 NHL games or fewer: Erik Karlsson (306 points, 400 games), Cale Makar (299 points, 280 games), Quinn Hughes (294 points, 328 games) and John Klingberg (275 points, 400 games). … Prior to the game, Buffalo recalled goalie Eric Comrie and assigned Devon Levi to Rochester of the American Hockey League.