The Lightning scored two goals in a 1:21 span to tie the game.
Gourde cut the deficit to 2-1 at 14:10 of the second period with a wrist shot from the right circle under the right arm of Andersen.
"I just thought we had a terrible start, you can't give that up," Tampa Bay coach Jon Cooper said. "We turned it into a 40-minute game and you're down 2-0 with two periods [left]. I thought we did a great job coming back and gave ourselves every chance to win that game."
D'Astous evened it 2-2, finishing a 2-on-1 rush when he tapped in Brayden Point's centering pass off Andersen's blocker as he crashed the net.
"Obviously they had a little push, but I thought after that we recovered really well," Andersen said. "And the third was a great example of that. Just got back to our game and [scored on] a little bit of a seeing-eye shot.
Logan Stankoven secured the 4-2 final with an empty-net goal at 18:29 of the third period after Andersen made a point-blank save on Nikita Kucherov's shot from in tight.
"Obviously started out great, and you get the goal to start the game," Brind'Amour said. "That gets you just, you know, feeling good about things. And I thought we were really good. We had a couple mistakes that they ended up capitalizing. But really, throughout the game, I thought we were solid. A couple big saves, a couple special plays from guys. Overall great effort."
NOTES: Aho has nine points (three goals, six assists) in his past nine games. ... Svechnikov has nine points (two goals, seven assists) in his past six games. ... Lightning forward Nick Paul and defenseman Erik Cernak each returned to the lineup. Paul, who missed 11 games with a lower-body injury, was minus-1 and had one shot on goal in 9:57 of ice time. Cernak missed two games with an upper-body injury and was minus-2 with one shot on goal in 16:01 of ice time. ... Gage Goncalves extended his point streak to three games (three goals, one assist).