"I think we've been playing well. We've found a rhythm and this is what we want to stick to in the future," Nylander said. "We were able to create a lot of chances and got two breakaways. I think we spent a lot of time in the offensive zone today so it was good."
Toronto is three points behind the Boston Bruins for second in the Atlantic Division, and six points behind the Lightning for first.
Kucherov scored to make it 3-2 after he gathered a loose puck off the stick of Killorn between the circles and shot past Andersen 58 seconds into the third period.
Gourde deflected a point shot from Brayden Coburn five-hole past Andersen 20 seconds later to tie it 3-3.
Video: TBL@TOR: Kucherov, Gourde strike 22 seconds apart
Van Riemsdyk broke the tie at 4:37 when he shot a rebound short side past Vasilevskiy from the left circle to make it 4-3.
"We have to play better each night because we play a team that's prepared for us and playing hard," Maple Leafs coach Mike Babcock said. "We've come a long way that way, but we're in a journey and we've got a long way to go. By no means are we a poised, veteran group who does it right. We still have these emotional swings in the game because we get off track. In saying all that, they're the best team in hockey and we can skate and play with anybody."
Nylander made it 1-0 at 11:55 of the first period when he took a pass from Matthews and one-timed a shot past Vasilevskiy's blocker.
Video: TBL@TOR: Nylander fires home goal from the circle
The Maple Leafs went up 2-0 at 3:08 of the second period when Matthews sent a lead pass through two defensemen to Nylander, who scored on a breakaway.
"They've got some good speed under their feet and can score goals, so we've got to do a better job eliminating that," Lightning defenseman Victor Hedman said. "Other than that, I thought we controlled the pace of the game. We created a lot of chances and it could have gone either way, but obviously, those guys are two top guys in the League, so next time we play them, we have to make sure we shut them down a little bit better."
Gardiner extended the lead to 3-0 at 11:36 when he took a pass from Nylander and shot from the high slot, using Lightning defenseman Andrej Sustr as a screen.
Video: TBL@TOR: Gardiner lasers home a nifty wrist shot
Killorn scored to make it 3-1 at 15:59 when he put a rebound past Andersen after he made the initial save on Brayden Point's shot.
"It's just another win. We expect to win every night, doesn't matter who we play," Matthews said. "Obviously, when you play the best team in the League, you have to bring your A-game, so I thought we played well tonight."
Goal of the game
Van Riemsdyk's goal at 4:37 of the third period.
Video: TBL@TOR: van Riemsdyk's shot sneaks by Vasilevskiy
Save of the game
Andersen's save on Tyler Johnson with 10 seconds left in the third period.
Video: TBL@TOR: Andersen denies Johnson twice in close
Highlight of the game
Nylander's goal at 3:08 of the second period.
Video: TBL@TOR: Nylander nets his second of the game
They said it
"I thought it was anyone's game. They came down and scored and that kind of takes the momentum away after you tie a game up like that. That obviously hurt us and we couldn't get one at the end." -- Lightning captain Steven Stamkos
"We knew they were going to come with a push and they did. They've got some good players over there so it's tough to keep guys like that off the board the whole night. It was a good response though by our team I thought." -- Maple Leafs forward James van Riemsdyk
Need to know
Patrick Marleau played in his 1,551st game, passing Alex Delvecchio for 13th in NHL history. ... Gourde has nine points (four goals, five assists) during a six-game point streak. ... Hedman played in his 600th NHL game, all with the Lightning. He has 338 points (73 goals, 265 assists) in his career. ... Nylander has seven points (four goals, three assists) in his past four games. ... Matthews has seven points (three goals, four assists) in his past four games.
Lightning: At the Buffalo Sabres on Tuesday (7 p.m. ET; MSG-B, SUN, NHL.TV)
Maple Leafs: Host the Columbus Blue Jackets on Wednesday (7 p.m. ET; NHLN, SN, TVAS, FS-O, NHL.TV)
Video: Nylander, van Riemsdyk lift Leafs over Bolts, 4-3