The home side were up by 20 points at the half courtesy of a pair of field goals by kicker Daniel Carlson and two touchdowns from quarterback Derek Carr, whose counterpart Murray then replied in the third quarter with a five-yard touchdown pass to Greg Dortch.
In the fourth, Murray ran in a two-point conversion after Arizona running back Darrel Williams scored from the one-yard line - before the signal-caller reloaded to rush for a touchdown, which he converted with a toss to wide receiver AJ Green for another two points to lock up the scores as time expired.
Las Vegas wide receiver Hunter Renfrow then fumbled the ball which was picked up by Cardinals cornerback Byron Murphy, who took it to the house for a 59-yard return and the win, prompting Murray to gleefully skip across the field and pump his throwing arm with joy.
Frustration got the better of Tampa Bay Buccaneer Tom Brady in New Orleans, with the quarterback throwing his electronic device in frustration during the third quarter against the Saints.
But the veteran took control, and his defence capitalised on three interceptions against New Orleans QB Jameis Winston, as Brady snapped a four-game regular-season losing streak against the Saints with a 20-10 result.
The sides' NFC South rivals the Carolina Panthers and the Atlanta Falcons suffered losses to the New York Giants and the Los Angeles Rams respectively, while the San Francisco 49ers shot to the top of the NFC West with a 27-7 defeat of divisional rivals the Seattle Seahawks.
In other games, the New York Jets just beat the Cleveland Browns 31-30, the Jacksonville Jaguars humbled the Indianapolis Colts 24-0, the Washington Commanders fell 36-27 to the Detroit Lions and the New England Patriots squeaked home 17-14 against the Pittsburgh Steelers.
Tua Tagovailoa turned in a career-best performance to lead his Miami Dolphins to a comeback road win over the Baltimore Ravens.
The young quarterback threw for 469 yards and six touchdowns in the 42-38 result, which saw him join Dan Marino as the only other player in franchise history to record at least 400 yards and five scores in a single game.
Running back David Montgomery had 122 yards on 15 carries but it was not enough for his Chicago Bears, who lost to the Green Bay Packers for the seventh consecutive time as they went down 27-10 in Wisconsin.
Russell Wilson failed to impress but crucially secured victory at home for the Denver Broncos against the Houston Texans, with the quarterback completing fewer than half of his 31 passes and being sacked three times in the 16-9 effort.
And the Dallas Cowboys held on to beat the Cincinnati Bengals 20-17, with Brett Maher kicking a 50-yard field goal as time expired.