Lucic had an assist and added in a fight for good measure, falling just short of the Gordie Howe hat-trick.
He was impressive all night, throwing his body around and controlling the puck along the boards.
Johnny Gaudreau was the hero for the visitors, scoring just 36 seconds into overtime with a lovely deke to tuck the puck around the outstretched right pad of Rittich for his seventh of the season as the Flames got two huge points in their push for a playoff berth.
On the play, Lindholm - who scored in the second period - settled down a bouncing puck and slid over a beautiful pass to set up Gaudreau.
Juuso Valimaki had the other goal for the Flames.
Jacob Markstrom got the win, making 24 saves, while Rittich had 25 stops.
Lucic's hard work along the wall set up the first goal, with the big man battling for possession before spying a wide-open Valimaki who had snuck in from the point, passing it cross-ice for an easy one-timer into an open net at 4:37.
It was Valimaki's second goal of the season.
It was also No. 17's 17th point of the campaign, and the 325th assist and 538th regular-season point of his career.
Lucic was recognized for his achievement during an early TV timeout with a video, standing and smiling with a wave to the 'crowd' as they came back from break.
The big man took a pass and rifled the puck high and wide hunting for a top corner not long after.
Brett Ritchie tipped a puck up by the Calgary point and showed off some good wheels for a partial breakaway, but Leafs blueliner Travis Dermott was able to catch him and knock the puck off his stick.
Toronto tied it up at 10:37 after a spirited one-on-one battle in the Flames zone between Sean Monahan and John Tavares. The Leafs centre was finally able to get the puck to Morgan Rielly, who snapped a wrister wide wide of the cage but it rebounded off the boards to the stick of Zach Hyman at the side of the net for a quick tuck up high.
Off the ensuing faceoff, Lucic dropped the mitts with Scott Sabourin, both players throwing a flury of punches but the Leaf ending up on the ice with Lucic standing over him. He did leave the ice for a quick repair after being cut during the melee.
According to Sportsnet Stats, Lucic became only the fifth player in league history to fight in his 1,000th game, joining Chris Neal, Donald Brashear, Shayne Corson and Geoff Courtnall.
Late in the first, Mikael Backlund tried to send a backhand pass across the crease to Andrew Mangiapane, but Rittich was quick to poke the puck out of harm's way before it found its target.
Mitch Marner was sent to the penalty box just 27 seconds into the second period for tripping Lindholm and the Flames forward made him pay, scoring his second powerplay goal in as many games just seven seconds later.
Lindholm won the faceoff leading to a tic-tac-toe play, Gaudreau feeding Matthew Tkachuk beside the net who one-touched it to Lindholm in the slot for a one-timer, similar to the tally he notched against the Oilers Saturday night.
Lucic then had Bobby Orr-esque skate up the ice later in the middle stanza, eventually splitting the Leafs D by going between Dermott and Zach Bogosian but losing the handle on the puck before he could shoot it.
Markstrom made an absolutely stellar, save-of-the-year third-period stop on Marner, raising his left pad up as he was laid out on the ice to stop an offering headed for an empty cage.
The Flames got hemmed in their zone and weren't able to clear a puck out, leading to Alex Kerfoot's goal to knot the game at 6:15.
The goal gave the homeside some energy, as they got some quality chances after that but Markstrom made a handful of big saves to deny them.
Toronto then got their first powerplay of the game - a four-minute high-sticking minor to Rasmus Andersson, who hit Joe Thornton in the face.
The Flames PK rose to the challenge, killing it off. The Leafs hit a post, but Mangiapane also had a great chance short-handed, Rittich making a five-hole stop.
Calgary took their turn to pour it on after that, Lindholm having two great chances on one shift - Rittich kicking out his left pad to deny a one-timer, then getting a piece of a wrister with the shaft of his stick.
"It just adds so much more to it. When we're desperate for wins and we get a big win like that against the best team in our division ... We fought hard all night long to get it. Kudos to my teammates and coaches for stepping up and getting the win to add onto the milestone."
"I just wish I didn't miss the net on the shift before the fight or else I would have had the Gordie in three shifts. It would have been nice, obviously, would have been a cool storyline. But personally, it's a great achievement and it makes it that much better to get the win."
Gaudreau on winning for Looch:
"Great night. Obviously, a great teammate, a guy that a lot of us in our locker-room look up to. To be part of his 1,000th game and get a win for him was a great night."
"The penalty-kill did a good job, the powerplay scored a big goal for us. That second period was real tight checking. We knew - they're the best team in the league - that they were going to push in the third period and they did and we hung on."