Phoenix came out with an intensity missing the previous two games, peppering Minnesota goalie Niklas Backstrom with 31 shots the first two periods. Backstrom was stellar at times, extending his scoreless streak to a franchise-record 157:13 before Pyatt finally broke through early in the third.
Sami Lepisto set up Pyatt's goal - Phoenix's first in 132 minutes - by keeping the puck in Minnesota's zone and Bryzgalov made it hold up with his 20th shutout in his 300th career game.
Backstrom, who set the previous scoreless record of 149:14 in 2008-09, turned away 41 shots.
This was a matchup of two teams heading in opposite directions.
Minnesota has been on a roll, particularly on the road.
The Wild, who won 13 road games all last season, came in 11-3-1 on the road since a 3-2 victory at Phoenix on Dec. 9 and had won six of seven overall after beating the Avalanche 4-3 Thursday night in Colorado.
Getting out to a fast start has been the key; Minnesota had outscored opponents 15-5 in the first period of its previous 12 road games and had the opening goal in all 15 of its wins away from St. Paul.
The Coyotes have simply been outworked over the past month, something they can't afford.
Bereft of a true star - captain Shane Doan is the closest thing - Phoenix is a team that grinds out wins with effort and good goaltending.
The Coyotes haven't gotten much of either lately.
One of the NHL's best goalies a year ago, Bryzgalov has been good, not great. His goals-against average of 2.68 is nearly half a goal more than last season and he had allowed at least four goals in five of his previous seven games.
One of the league's hardest-working teams last season, Phoenix also has frustrated coach Dave Tippett with sporadic effort. The Coyotes blew a two-goal lead to last-place Edmonton just before the All-Star break, lost a three-goal lead to San Jose on Tuesday and got walloped 6-0 by Vancouver at home on Thursday.
Effort wasn't a problem early on for the Coyotes.
Phoenix fired 16 shots at him and had numerous good chances in the opening period, only to get turned away each time. Backstrom made multiple saves on three separate flurries, capping it with a spectacular kick save on a point-blank shot by Martin Hanzal in the closing seconds.
Backstrom turned away 15 more shots in the second period - two on snatching glove saves - and Bryzgalov held his ground, too, giving the Coyotes a chance.
Pyatt came through 2:20 into the third, one-timing a shot from the slot after Lepisto swiped a puck away from Minnesota's blue line to keep it in the zone for Phoenix's second win its past six games.
Notes: Phoenix D Ed Jovanovski missed his second straight game with an upper-body injury. ... The Wild are 4-18-1 when allowing the first goal. ... The Coyotes had allowed 11 straight goals before Pyatt scored.
(This version CORRECTS Coyotes 1, Wild 0. Corrects Bryzgalov's career shutouts to 20.)