Jonathan Dahlen and Tomas Hertl each had a goal and an assist, and Adin Hill made 22 saves for the Sharks (14-12-1), who have lost three of their past four.
"It wasn't a great game," Hertl said. "I don't think we had our legs tonight. If you don't have your legs you have to find a way, play simple hockey and don't turn over pucks."
Fiala scored his first goal in eight games to make it 1-0 at 6:22 of the first period. Hill did not recover in time from stopping a shot through traffic by Jon Merrill, and Fiala scored on the rebound.
"On that first goal, we didn't get inside at the net and that's one of the things we talked about in our scouting report on [Minnesota]," Sharks coach Bob Boughner said. "They're heavy around the offensive net and around the blue [paint] and you've got to get inside. You've got to be underneath their sticks and we weren't."
Eriksson Ek made it 2-0 with a power-play goal at 7:04 of the second period when he redirected a shot from Mats Zuccarello. Kaprizov had the secondary assist to extend his point streak to seven games (five goals, nine assists).
Greenway scored another on the power play at 15:20 with a wrist shot off of the rush to extend the lead to 3-0.
"I haven't been on the power play a ton, so it's good, first to have the coaches trust me to be out there, and then take advantage of the opportunity like that," Greenway said.