Dalton completed 34 of 54 passes for 266 yards, one touchdown and two interceptions in the 38-10 home loss at AT&T Stadium. He was forced into action after starter Dak Prescott sustained a season-ending ankle injury a week ago.
"There are some good things that we can take away, but there are a lot of areas that we need to improve," Dalton told reporters.
"I think it's one of those things where everybody has to look at themselves. I've got to be better. I've got to do a better job out there and giving our guys chances."
The Cowboys had four turnovers, allowed three sacks and totaled just 344 yards in the loss. Pro Bowl running back Ezekiel Elliott fumbled twice and ran for just 49 yards.
Cardinals running back Kenyan Drake had 20 carries for 164 yards and two scores. Strong safety Budda Baker had seven tackles, an interception, a sack and a forced fumble in the win.
"He is everywhere," Cardinals coach Kliff Kingsbury said of Baker. "You watch the game, and No. 32 shows up in every situation, all night. He is such a tremendous player."
Both offenses had sluggish starts as the game began with five consecutive possessions that ended in punts. Elliott lost his first fumble near the end of the first quarter.
The Cardinals responded with an 11-play, 54-yard scoring drive punctuated by a six-yard touchdown pass from quarterback Kyler Murray to Christian Kirk to give Arizona a 7-0 edge.
Elliott lost another fumble on the Cowboys next drive and the Cardinals took over at the Dallas 26-yard line. Drake ran in for a one-yard score five plays later to double Arizona's lead.
After the Cowboys offense was again forced to punt on their next possession, Murray again found Kirk for an 80-yard touchdown toss to grow the Cardinals lead to 21-0.
Cowboys kicker Greg Zuerlein made a 34-yard field goal just before the half to get Dallas on the board, 21-3 at the break.
Dalton threw an interception on the first drive of the second half before Murray orchestrated a nine-play, 73-yard touchdown drive that ended with a one-yard rushing score.
Dalton threw his second interception of the day 1:40 into the fourth quarter, and Cardinals kicker Zane Gonzalez made a 26-yard field goal on Arizona's possession for a 31-3 edge.
The Cowboys scored their lone touchdown on the next possession, as Dalton conducted an 18-play, 62-yard drive that ended with a one-yard touchdown toss to Amari Cooper.
Drake ripped off a 69-yard touchdown run for Arizona three plays later for the final points of the game.
"It was just exciting to see our team play on a Monday Night Football stage," Cardinals coach Kliff Kingsbury said. "The way they embraced it with effort and energy was meaningful to me."
Murray completed just 9 of 24 passes for 188 yards and two touchdowns in the win, but also had 74 rushing yards. Cardinals linebacker Haason Reddick had two sacks and a pass defensed.
The Cowboys (2-4) hit the road to face the Washington Football Team at 1 p.m. EDT on Sunday at FedEx Field, and the Cardinals (4-2) host the Seattle Seahawks at 4:05 p.m. EDT at State Farm Stadium.