"I might not have told the whole truth this morning when I said I didn't want to win this game that bad," Smith said.
But it was clear what this game meant to Smith from the massive fist-pump he let out with after stopping Ryan Callahan's shootout attempt to end the game.
Smith, who made 28 saves, was the Lightning's goalie from 2007-2011. His tenure ended in disappointment and a stint in the minors.
"You get in the game and I've got friends and family here watching the game too and it was my first game back here and there was definitely extra motivation to play well tonight," Smith said.
Ondrej Palat scored his second goal of the game with 38 seconds remaining in regulation to tie the game at 3-3 and send it to overtime after Mikkel Boedker gathered a loose puck and backhanded it past Lightning goalie Ben Bishop to give the Coyotes the lead at 7:48.
"That was real important for my confidence," Boedker, who hadn't scored since Jan. 24, said. "It feels good. It was coming for me tonight and that always feels good."
Tampa Bay began the second and third periods with a one-goal lead, but Phoenix equaled the score early in each period.
"We tried to play fast tonight; we put some young legs in the lineup tonight and they play fast," Phoenix coach Dave Tippett said. "We are desperate for points. You never know how that will turn out, but it turned out alright for us."
Erat, acquired prior to the NHL Trade Deadline from the Washington Capitals, scored his second of the season 2:13 into the third period, when Victor Hedman's attempt to bounce the puck off the end boards behind the Lightning net failed. The puck slipped in front of the crease, where Erat sent it behind Bishop, who was looking for the puck to his right. The goal tied the score at 2-2.
"We don't [follow] the D-to-D pass behind the net and the goaltender follows the puck. We're way too lackadaisical and it's a mental error," Lightning coach Jon Cooper said. "That ends up in the net. The third goal, we've run that coverage a thousand times. Just a blown assignment. That shouldn't happen in midget hockey, let alone the NHL. That's what frustrating. The guys played hard, but we made some mental errors down the stretch. That's the frustrating thing."
Brandon McMillan scored his second goal in as many games when he sent a rebound under Bishop's right pad to make it 1-1 at 1:01 of the second period.
The tie score held up for most of the period until Palat scored his 16th of the season with 53 seconds remaining in the period. Callahan dug the puck out from behind the Phoenix goal, cut in front, got a shot away that Smith blocked, but then edged the puck over to Palat. Keith Aulie also had an assist on the goal.
Tyler Johnson scored his NHL-leading fourth shorthanded goal of the season and 20th overall at 11:26 of the opening period to give Tampa Bay a 1-0 lead. Johnson took a pass from Palat, cut in front of the Phoenix net and backhanded the puck over Smith.
Phoenix went 0-for-3 on the power play. The Lightning have not allowed a power-play goal in their past six home games.
Bishop stopped 17 shots for Tampa Bay. With the loss, the Lightning's shootout record for this season fell to 6-5.
"The well has gone dry for us in the shootout," Cooper said. "That's something we used to do pretty well at. Just another thing that's tough. That's 0-for-10 in shootouts in 48 hours. I don't have the answer. I would never have suspected the results we are getting."
By contrast, the win sent the Coyotes, in the midst of a four-game road swing, out of town on a hopeful note. Phoenix will visit the Florida Panthers on Tuesday.
"When things get going the right way, those are things that fall into place," Tippett said. "We haven't had enough of those this year. Hopefully this is a sign that there are good things to come."
Lightning defenseman Sami Salo left in the second period with a lower-body injury and did not return.