Brad Stuver had two saves for The Verde who followed the first win in franchise history, a 3-1 victory at Colorado last Saturday, with their second straight road win.
Dayne St. Clair finished with three saves for Minnesota United, who began their season with three consecutive losses for the first time in franchise history.
The Loons, who scored a total of one goal in its first two losses to Seattle and Real Salt Lake, missed a good chance to take an early 1-0 lead in the 10th minute when Stuver left his line and weakly punched a ball directly to a wide-open Ethan Finlay on the right side of the box. But Finlay's chip shot at the open goal sailed high over the net.
Austin FC took a 1-0 lead seven minutes later when Jared Stroud curled a crossing pass from the right side of the box perfectly between defender Romain Metanire and St. Clair to Fagundez, who easily tapped it into the left corner of the net for his second goal of the season.
Minnesota nearly tied it in the 31st minute when midfielder Emanuel Reynoso weaved his way through several Austin defenders and fired a left-footed shot from the middle of the box that caromed off the left post and straight out.
The Verde then missed a chance to make it 2-0 just before halftime when Stroud curled another pass from the right side of the box to Danny Hoesen on the far post, but Hoesen's left-footed attempt was high and wide of the goal.
Minnesota's best chance to tie it in the second half came in the 83rd minute when 17-year-old forward Patrick Weah, the 2017 Minnesota State Player of the Year who was making his MLS debut and is the cousin of U.S. National Team player Tim Weah, headed Justin McMaster's crossing pass straight into Stuver's chest from close range.