Pat Maroon also scored for the Lightning, who improved to 5-0 at Amalie Arena. Andrei Vasilevskiy finished with 27 saves.
Goodrow's second goal of the game was the 6,000th goal in Lightning franchise history.
Tampa Bay extended its point streak against Detroit to 19 games (18-0-1) and has won 16 consecutive regular-season home games against the Red Wings.
``We didn't have the right approach, we didn't play well,'' Lightning defenceman Mikhail Sergachev said of the team's effort. ``We gave them a lot of scoring chances and made it easy for them. But I thought we did a really good job in the third period, the boys just dialed in and shut them down.''
Vladislav Namestnikov scored for Detroit, which is winless in eight games (0-6-2). Thomas Greiss finished with 29 saves.
The Red Wings showed a more concerted effort in the rematch against the defending Stanley Cup champions and were rewarded with the opening goal of the game halfway through the first period.
Namestnikov, a former Lightning first-round pick in 2011, came out from behind the net and established position in front as Filip Zadina shot from the left circle. The shot deflected of Namestnikov and past Vasilevskiy at 10:00.
``We controlled what we can control and we showed up,'' Detroit captain Dylan Larkin said. ``We competed. That's a heck of a hockey team over there. It was a one-shot game. We wanted to clean up some small things. We were right there. We worked the way we should every night.''
Tampa Bay tied it at 13:56 as Maroon intercepted a pass curling back in the offensive zone and wheeled to the high slot where he sent a wrist shot through an Alex Killorn screen and past Greiss for his first of the season.
Detroit dominated most of the second period, but it was Tampa Bay that scored the only goal of the frame as Jan Rutta touched a pass back to Goodrow coming off the bench before knuckling a one-timer that hovered just off the ice and through the legs of Greiss, who was screened by Steven Stamkos at 17:29.
``A bad, bad second period, I thought that was our worst period of the season,'' Maroon said. ``They just did a really good job of clogging it up and make it a frustrating night on us, but we made it harder on ourselves also by turning pucks over at the blue line and feeding into their game. So we have to be better on that end, but I thought Detroit played really well tonight.''
Goodrow secured the win with his second of the night from centre ice into the open net with 27.3 seconds left for his first career multi-goal game.
``That's more conducive to the type of hockey we have to play to be successful,'' Detroit head coach Jeff Blashill said. ``They leaned on us a bit in the third but up until that point we hadn't given them much. We're going to be in those 2-1 games and we have to find ways to win them. It's playing within the structure and playing the right way.''
DETROIT INJURIES: Detroit RW Filip Zadina returned to the lineup after missing seven games while on the league's COVID list. C Valtteri Filppula was a scratch, though the team did not offer an explanation for his absence.
ON THE BOARD: Maroon's goal was his first point of the season, the last of Tampa Bay's 20 skaters to suit up this season to record a point.
Detroit Red Wings: At Florida Panthers on Saturday.
Tampa Bay Lightning: At Nashville Predators on Monday.