Matthews gave the Maple Leafs a 2-1 lead at 1:46 of the second period with a snap shot from the right circle, but Rutta tied it 2-2 at 5:46 with a backhand from the slot.
Matthews scored again at 9:57 with a redirection from Marner to give Toronto a 3-2 lead, and Alex Kerfoot made it 4-2 at 12:25.
"This is a big game for us, obviously," Matthews said. "They're a really good team, they know how to win, so it was an important game for us this point of the season. All these points count and really matter. I think it's just a really complete game by everybody."
Matthews completed the hat trick at 7:08 of the third period with a shot from the low slot to make it 5-2, and Marner scored on a one-timer at 11:24 for the 6-2 final.
"Can't give a guy like him time and space the way we did," Cooper said. "You're leaving a 50-plus goal scorer all alone at the net and he's walking right down Broadway and nobody is touching him. That's on us."
NOTES: Matthews extended his point streak to 13 games (17 goals, seven assists). He is the fifth player in Maple Leafs history with at least four hat tricks in a season (Reg Noble, 1917-18; Babe Dye, 1922-23, 1924-25; Darryl Sittler, 1980-81; Vaive, 1983-84). ... Marner has 15 points (two goals, 13 assists) during a six-game multipoint streak and 22 points (six goals, 16 assists) during a 10-game point streak. ... Toronto forward William Nylander did not play because of an illness. ... Stamkos extended his point streak to seven games (five goals, six assists); Kucherov has 10 points (two goals, eight assists) during a five-game point streak. ... Rutta's goal was his first since Nov. 20 against the New Jersey Devils. … Lightning defenseman Ryan McDonagh participated in the morning skate but missed his sixth straight game because of an upper-body injury.