Andrei Vasilevskiy made 27 saves for Tampa Bay, which played without suspended forward Nikita Kucherov and injured defenseman Victor Hedman (undisclosed).
Kucherov, who led the NHL with 128 points (41 goals, 87 assists) in the regular season, was suspended one game by the NHL Department of Player Safety for boarding Blue Jackets defenseman Markus Nutivaara at 15:34 of third period in Game 2 on Friday.
Nutivaara (upper body) did not play Sunday.
"[Kucherov]is a big part of our team," Lightning coach Jon Cooper said. "The leading scorer in the League. He's definitely going to be an MVP finalist, so anytime you take someone like that out of the lineup, as well as Hedman, but I thought we battled."
Duchene gave Columbus a 1-0 lead at 1:44 of the second period, getting the rebound in the right circle after Zach Werenski's initial shot was deflected in front by Cam Atkinson off Andrei Vasilevskiy's left pad.
Oliver Bjorkstrand extended the lead to 2-0 at 8:25 on the power play, taking a pass from Seth Jones and scoring from the top of the right circle. Alexandre Texier had the secondary assist for his first playoff point.
"We played a patient game, each team had surges," Jones said. "The power play got us the second goal. A lot of momentum swings. I thought we handled ourselves very well tonight.
"They surged on us in the third. They started making plays and accelerating through the neutral zone, but we hung in there."
Tampa Bay outshot Columbus 17-8 in the third period.
"We started to make some plays," Lightning forward Tyler Johnson said "Throughout the game and throughout the series, I thought we were keeping it too simple in a way. We have a lot of skill in this room, we need to use it."
Ondrej Palat cut it to 2-1 at 4:40 of the third period after getting his own rebound and beating Sergei Bobrovsky, who made 30 saves, five-hole from in front.
Atkinson scored into an empty net at 19:00 for the 3-1 final.
"If anything, maybe we found a recipe to break some of their structure there, but obviously it wasn't good enough," Lightning center Steven Stamkos said.