The Lightning let a 3-1 lead slip away to the Blackhawks, who improved to 17-11-3 on the road with 37 points, fourth-best in the league. Chicago's 95 goals away from home are second only to Detroit's 102.
Dave Bolland scored the go-ahead goal, his 13th of the season, with 7:14 left in the third period, and Toews added his second goal of the game at the 16-minute mark to make it 5-3.
"We stuck with it," Blackhawks coach Joel Quenneville said. "Obviously we got a huge goal [at the end of the second period] and that ignited us into the third. I like the way we're playing. We didn't change much at that point -- we haven't at all on the road -- and we just kept coming like we needed to."
The Lightning tied the game 1-1 just 1:05 later when Stamkos scored his eighth of the season.
Stamkos, the No. 1 overall draft pick last summer, scored twice in the second period, as a smattering of hats flew to the ice.
"This was one of the best feelings of the year for me so far, but I much rather have it in a win," Stamkos said.
The 19-year-old, who had just six goals in his first 53 games, now has four in his past three and became the first rookie in franchise history to record a hat trick.
"It's big because he knows he can play against a quality team like that," Lightning interim coach Rick Tocchet said. "It's a measuring stick for him."
Stamkos, who had struggled because of lack of playing time, says he feels more confident on the ice.
"I'm the biggest critic of my game," Stamkos said. "At the beginning of the year, with all the stuff that was going on, playing five, six, seven minutes a game, it's tough to get confidence."
Stamkos scored on a double power play in the second, when he put a wrist shot into the top corner. His third came just 1:10 later on a one-timer past Cristobal Huet on the short side. Vaclav Prospal picked up assists on both goals, giving him 23 on the season.
"They have some great players," Kane said. "Like Stamkos. You can obviously see the talent he has. Overall, though, I think we did a good job of shutting plenty of people down, and you can tell we pretty much dominated."
Chicago pulled to 3-2 late in the second with a goal from Toews, assisted by Kane.
The Blackhawks tied the game 1:46 into the third on Kane's 18th of the season. Kane leads the team with 51 points.
Ramo, in his first start since Feb. 3, made a career-high 44 saves on 49 shots. Huet made 19 saves for Chicago.
"Tonight, I took another step [forward]," Ramo said. "I didn't finish well, but you have to be able to pick up the good parts and build on that."
Patrick Sharp, Chicago's leading goal scorer with 23 goals, will be sidelined indefinitely with a left knee injury. Quenneville expects Sharp to be out at least a couple weeks. ... Nikolai Khabibulin did not make the trip to Tampa after aggravating a groin injury at Monday's practice. The former Lightning goalie, who helped Tampa Bay win the 2004 Stanley Cup, has yet to play his former team since signing a four-year, $27 million deal with the Blackhawks. ... Tampa Bay defenseman Marek Malik missed Tuesday's game with a lower body injury he sustained in Saturday's game against the Washington Capitals.