Rinne made 22 saves for his NHL-leading 21st win. It was Ekholm's first goal since a 3-2 victory against San Jose on Jan. 7.
"Great goal for him, but a big one for our team as well," coach Peter Laviolette said.
Ekholm circled from behind the net and flipped a pretty wrist shot past Niklas Backstrom. The nice move silenced the crowd, which just moments earlier erupted when Thomas Vanek sent the game to overtime with 48 seconds to play.
"I was just kind of looking for the D, then I saw no one was there so I was just trying to spin in front of the net and got a shot off," Ekholm said. "It wasn't the hardest, but it went in."
Zach Parise scored two goals for Minnesota, and Backstrom finished with 27 saves after replacing starter Darcy Kuemper, who had a rough first night back after missing a game with an illness. Kuemper allowed three first-period goals on 14 shots and was pulled for the third time in his last four home starts.
"You score five goals, for our team it should be automatic," Yeo said.
Nashville led 3-2 after Smith backhanded a shot off Kuemper's glove and into the net with just over a minute to play in the opening period. But the Wild tied it at 7:18 of the second when officials awarded Marco Scandella a power-play goal after a lengthy review.
With Rinne facing the net, Scandella's attempt off a rebound appeared to sneak under Derek Roy and defenseman Seth Jones, who was behind Rinne in the net.
"I thought I had the puck when they blew the whistle," Rinne said. "But I'm not going to complain really. I haven't seen enough reviews."
After several minutes, referee Justin St. Pierre announced Scandella had been awarded the goal, and the NHL released a statement saying the review showed the puck "completely crossed the goal line."
NOTES: Minnesota is the only team to score five goals against Rinne this season. ... Cullen scored his first goal against his former teammates. The Minnesota native played three seasons with the Wild from 2010-2013, recording 101 points (33 goals, 68 assists). ... Wild LW Matt Cooke returned to the lineup after missing 22 games with a lower-body injury. C Ryan Carter (shoulder) also returned after missing three games, although the Wild lost C Mikael Granlund, who missed the game with an illness. ... Smith has five goals and six assists in 13 career games against Minnesota.