Coleman made it 2-0 on a power play at 13:48, taking a pass from Bobby Brink and going backhand-forehand over Saros' right pad from in tight.
Coleman had been acquired from the Calgary Flames along with defenseman Olli Maatta for three NHL draft picks and defenseman Jake Middleton on July 2.
"It's nice to see new guys come in and, I think, get their footing right away and help the team win a game," Hynes said.
Yakov Trenin appeared to extend the lead to 3-0 with a shot in off the left post at 15:05, but the goal was overturned after the Predators successfully challenged that Ryan Hartman interfered with Saros.
"It was a lot of sloppy, sloppy puck play from our side," Nashville forward Filip Forsberg said. "Obviously, they got those two goals pretty early and then it was a little bit uphill from there."
Josi cut the deficit to 2-1 at 9:23 of the second period. Forsberg sent the puck into the right face-off circle, where Josi one-timed it from a sharp angle off Wallstedt's glove just three seconds after a power play expired.
"We'll look at some things tomorrow, work on some things we can do better," Josi said. "We've got to wash this tomorrow after we look at it and focus on Saturday (against the Dallas Stars). ... I don't think it was a lack of urgency. I think we made some mistakes that we have to get out of our game, but I thought the urgency was there."
Coleman had a chance to score again after taking a pass from Matt Boldy on a 2-on-0, but was denied by a pad save from Saros at 11:33.
Kaprizov pushed the lead to 3-1 at 4:42 of the third period by tipping a point shot from Quinn Hughes.
"I thought, overall, it was a pretty solid game for us," Hynes said. "Nashville made the push and scored right at the end of that power play. I thought we managed the game well in the second period and then I really liked our third period."
NOTES: With an assist on Josi's goal, Forsberg has at least one point in each of his past 11 season openers (15 points; five goals, 10 assists), tying Vincent Lecavalier (11 games from 2002-03 to 2013-14) and Viktor Kozlov (11 games from 1997-98 to 2008-09) for the longest streak in NHL history. ... Kaprizov has at least one point in each of his seven season openers since entering the NHL in 2020-21 (11 points; two goals, nine assists). ... Defenseman Jonas Brodin tied Mikko Koivu (15) for the second-most seasons played for the Wild behind defenseman Jared Spurgeon, who extended his record to 17 seasons. ... Hughes had his 10th assist in a season-opening game to tie John Carlson and Mark Howe for the fifth-most by a United States-born defenseman in NHL history, behind Gary Suter (16), Ryan Suter (12), Phil Housley (11) and Justin Faulk (11).