Morey rushed untouched off the left side of the Dallas line to block the kick. Beisel, a backup linebacker, picked up the ball at about the 3 and bowled over the goal line.
McBriar was hurt on the play and had to be carted off the field in the din of a crowd that included the usual large portion of Dallas fans.
The stunning play by two of Arizona's most unheralded players brought a conclusion to a game that began with a 93-yard kickoff return by the Cardinals' J.J. Arrington followed by many moments of confusion, controversy and clutch performances.
Dallas (4-2) scored 10 points in the final 2 minutes of regulation, sending the game into overtime when Nick Folk's 52-yard field goal barely cleared the crossbar as the fourth quarter ended.
Folk was in field-goal range only because a 5-yard offside penalty was called against injured Arizona linebacker Travis LaBoy as he tried to limp off the field. That moved the ball from the 40 to the 35.
That was just one of the weird moments in Arizona's sixth consecutive home victory.
Arizona (4-2) scored 17 consecutive points, including a pair of touchdown passes by Kurt Warner, to go up 24-14 on Neil Rackers' 40-yard field goal with 3:17 left in regulation.
Tony Romo, who threw three touchdown passes but fumbled twice, connected with Marion Barber on a 70-yard scoring play that cut it to 24-21 with 2 minutes left.
Dallas' defense held, and the Cowboys got the ball at their 32. Romo connected with Patrick Crayton on a 30-yard play to the Arizona 39, where the Dallas quarterback spiked the ball with 4 seconds to play to stop the clock.
LaBoy was far downfield, and was called offside.
Officials, who had a difficult day all around, huddled for several minutes and even reviewed the play before Folk's game-tying kick.