Landeskog's third-period goal erased the one-goal lead Trevor Daley gave the Penguins 9:12 into the third.
After Evgeni Malkin's apparent goal was waved off 7:07 into the third period following a coach's challenge, Daley gave the Penguins a 3-2 lead on Pittsburgh's second power-play goal. Matt Cullen set up Daley for a slap shot he sent by goalie Calvin Pickard , who made 28 saves in his first start of the season.
The Avalanche won the first game of a four-game road trip after defeating the Dallas Stars 6-5 in its season opener Saturday.
The Penguins won two of three games on their season-opening homestand despite being without center Sidney Crosby (concussion).
Phil Kessel scored his second power-play goal in as many games to open the scoring 8:57 into the first period. He shot past defensemen Francois Beauchemin and Erik Johnson before beating Pickard through the five-hole.
Cullen extended the lead 36 seconds later. After the puck slid out of the corner, Cullen retrieved it and drove to the net before shooting off Pickard, collecting his rebound and sending a wrist shot into the left side of the net.
With the goal, Cullen is the fourth-oldest player to score a point for the Penguins (39 years, 350 days).
"[Cullen] just plays the game the right way, for me," Penguins coach Mike Sullivan said. "He plays hard. He's a smart player. We can use him in all situations."
Fellow 39-year-old Jarome Iginla cut the Penguins lead in half with 3:33 remaining in the first period. The Avalanche went on a 5-on-3 power play for 1:11 after Penguins forward Chris Kunitz hooked Nathan MacKinnon with Nick Bonino already in the penalty box.
Iginla scored from the right circle 1:23 before Patrick Wiercioch tied it 2-2.
Goal of the game
After tying it with 6:32 left in the third period, Landeskog struck again early in overtime. MacKinnon passed off of Penguins defenseman Kris Letang's stick, causing the puck to flip in the air. Landeskog tracked it and whacked it past Fleury for a goal that he described as lucky.
Video: COL@PIT: Landeskog tips home game-winner in overtime
Saves of the game
Fleury made a series of saves late on a Colorado 5-on-3 power play in the third period to keep it tied 2-2. MacKinnon tipped the puck toward Pittsburgh's crease after Fleury fell forward but was denied twice. The puck came loose to Matt Duchene , but Fleury fought to gain possession and freeze it 5:44 into the third period.
Highlight of the game
With the Avalanche trailing 3-2 and 6:32 remaining in the third period, Landeskog tied it on the power play. Mikhail Grigorenko found Landeskog in the right circle, where he shot it by Penguins defenseman Ian Cole and caught Fleury leaning off his post.
Unsung moment of the game
Avalanche coach Jared Bednar challenged after Malkin seemed to give the Penguins a 3-2 lead 7:07 into the third. Malkin skated through the crease with Colorado defenseman Fedor Tyutin falling behind him, collected a shot-pass from Kessel that ricocheted off the end boards and shot behind Pickard and by Tyutin. But following a video review, it was ruled Malkin interfered with Pickard.
They said it
"The scary thing was that it was a couple seconds until they shot it in the net, but I believe it did affect how I would've played it." -- Avalanche goalie Calvin Pickard on Penguins forward Evgeni Malkin's disallowed goal
"I think we had some guys who were horses for us tonight and carried the rest of the guys on our team." -- Avalanche coach Jared Bednar on Gabriel Landeskog and Nathan MacKinnon
"I just wish we would have won that one. We were so close. It was disappointing." -- Penguins goalie Marc-Andre Fleury
Need to know
Crosby skated on his own before Pittsburgh's morning skate. … Avalanche forward Rene Bourque played after sitting out Colorado's practice Sunday.
Avalanche: At the Washington Capitals on Tuesday (5 p.m. MT; CSN-DC, ALT, NHL.TV)
Penguins: At the Montreal Canadiens on Tuesday (7:30 p.m. MT; TVA Sports, SN360, ROOT, NHL.TV)