Brayden Point had a goal and an assist for the Lightning, the No. 3 seed who won the first two games. Andrei Vasilevskiy made 24 saves.
Game 4 of the best-of-7 series will be played here Saturday (4 p.m. ET; USA, SN, TVAS).
"We've been in spots like this before," Point said. "We played a good game. I think it might be different if we felt like we didn't play a good game and they won. For the most part, I liked our game, I liked our compete. Give Carolina credit, they played hard as well. They got this one. We've got to be sharp for the next one."
Brett Pesce gave Carolina a 1-0 lead, its first of the series, with a one-timer from the top of the right circle at 5:15 of the second period.
"I'm just so proud of everyone on our team," Pesce said. "We get punched in the face over and over again ... tough bounces ... and we just stick with it. Kudos to our team and [Brind'Amour] to keep on relaying that message. We just kept at it and the power play came up big for us."
Aho extended the lead to 2-0 at 7:40 when he got behind the defense and shot five-hole.
"It feels great, but at the same time, this is our first win in the series," Aho said. "We just have to keep going and going for Game 4 and just do it over and over again and trust our system."
Point cut it to 2-1 with a power-play goal at 8:57, scoring off a backdoor pass from Nikita Kucherov.
"I think we were pushing the play, I think we were a lot better offensively," Tampa Bay defenseman Victor Hedman said. "We were winning our battles, creating some havoc. There were a lot of things we did well, it just didn't go our way tonight. If we bring the same mentality into the next game, I like our chances."