Veteran Brissett, filling in for starter Kyler Murray, punished a woeful performance by the Cowboys, who crashed to their first home loss of the season with a defeat that leaves them with a 3-5-1 record on the season.
"Just a complete team win," Brissett said after Arizona's third win of the season. "We knew it was going to be a dogfight coming in here. I thought we did a good job."
Brissett finished with 261 passing yards from 21 completions, and also rushed for a touchdown as a string of blunders from the Cowboys proved decisive at AT&T Stadium.
Cowboys quarterback Dak Prescott was sacked five times and Dallas coughed up four fumbles on their way to a damaging defeat.
The Cowboys had gotten off to an uninspiring start, with Prescott being sacked as the team attempted a fourth-down conversion near the Cardinals' goal line on the opening drive.
The Cardinals made the Cowboys pay for that miss, with Chad Ryland's 48-yard field goal giving the visitors an early 3-0 lead.
After forcing Dallas to punt, Arizona extended their advantage to 10-0 when Brissett connected with Marvin Harrison Jr. on a four-yard pass for a touchdown.
The Cowboys' flat opening continued soon afterwards, when tight end Jake Ferguson fumbled to surrender possession to the Cardinals once again.
But Dallas earned a badly needed breakthrough on the next Cardinals drive when Pat O'Donnell's punt was blocked by Sam Williams, allowing Marshawn Kneeland to snatch a touchdown.
Arizona restored their 10-point advantage just before halftime with Brissett leading an 11-play, 74-yard drive before burrowing over from close range for a touchdown.
The Cardinals kept up the pressure after halftime, with Brissett finding Trey McBride in the end zone to make it 24-7.
Dallas attempted to haul themselves back into contention, but another promising drive faltered with Prescott sacked again, forcing the Cowboys to settle for a field goal that made it 24-10.
Another Cardinals field goal put the Cardinals 27-10 ahead, and although the Cowboys cut the deficit to 10 points when Prescott completed a short pass to Ryan Flournoy in the fourth quarter, that was as good as it got for Dallas.
A fumble on their next possession left the Cowboys needing a miracle as the clock ticked down, and Prescott then tossed a desperate interception inside the final two minutes to seal his team's defeat.