Victor Hedman also scored for the Lightning, and Steven Stamkos had two assists.
Kucherov's 30th multi-point game this season gave him 93 points for the season. Edmonton's Connor McDavid, Pittsburgh's Evgeni Malkin and Colorado's Nathan MacKinnon are tied for second with 89.
Ty Rattie scored in his second consecutive game for the Oilers.
In a battle of backup goaltenders, Tampa Bay's Louis Domingue recorded 29 saves, and Al Montoya had 37 for Edmonton.
The win broke the Lightning's two-game losing streak. Tampa Bay earned its eighth victory in the past nine home meetings with the Oilers, including six straight.
Tampa Bay improved to 26-8-2 at home.
Both clubs were coming off the first game of back-to-back contests, with both games taking place in the Sunshine State on Saturday. Edmonton won 4-2 against the Panthers in Sunrise behind McDavid's three-point effort. Boston shut out Tampa Bay 3-0 behind goalie Tuukka Rask.
Domingue's puck-handling mistake while out of position on the game's first shift led to Rattie grabbing the puck behind the Lightning goal.
Rattie then fired a shot that ricocheted in off the netminder's right skate just 14 seconds into the game as Domingue tried to slide back into position.
Rattie's goal was his second in four games for Edmonton after being called up from the club's AHL affiliate in Bakersfield, Calif.
He had an empty-net tally against the Panthers.
Hedman evened it at 1-1 at 15:12 of the first period. His long shot from the point hit Rattie's stick, took an odd bounce and hopped past Montoya for his 12th goal.
Kucherov recorded his seventh power-play goal when he took a pass from Stamkos and snapped a wrist shot over Montoya at 2:24 of the second period.
The right winger added his 36th goal of the season during a scrum in front of the net at 11:34 of the third period on a weak shot that slipped past Montoya.
--Field Level Media
Edmonton | Tampa Bay | |
Ty Rattie 1 | Points | Nikita Kucherov 2 |
Ty Rattie 1 | Goals | Nikita Kucherov 2 |
Adam Larsson 1 | Assists | Steven Stamkos 2 |
N/A | Power Play Goals | Nikita Kucherov 1 |
N/A | Short Handed Goals | N/A |
Al Montoya .925 | Save Percentage | Louis Domingue .967 |
Al Montoya 37 | Saves | Louis Domingue 29 |
Team | Shots | Goals | Power Play | Penalty Kill | Penalty Mins | Face Offs Won |
Edmonton | 30 | 1 | 0-5 | 3-4 | 8 | 23 |
Tampa Bay | 40 | 3 | 1-4 | 5-5 | 10 | 29 |