In the other early game on Wednesday, Shakhtar Donetsk playmaker Heorhiy Sudakov had a fine assist and an even better goal in a 2-1 win for the Ukrainian champions against Young Boys.
Villa had led the 36-team Champions League standings after three straight wins without conceding a goal - and it took a bizarre incident before goalkeeper Emi Martinez was eventually beaten.
Villa defender Tyrone Mings was punished for picking up the ball when Martinez seemed to restart play with a goal kick passed short to his teammate.
Mings walked forward a couple steps to gather the ball with his left hand and returned to place it at the corner of the six-yard box.
The penalty was awarded and Brugge captain Hans Vanaken scored in the 52nd minute, placing the ball to Martinez's left as the World Cup-winning Argentina goalkeeper dived to his right.Â
Martinez tried to persuade German referee Tobias Stieler after the game that he had made a mistake.
The loss leaves Villa in fifth place in the standings now led by Liverpool, with Inter Milan, Arsenal and Brest all able to rise above coach Unai Emery's team with a win later on Wednesday. Inter were hosting Arsenal at San Siro.
Club Brugge have six points, from two wins and two losses.
Shakhtar's Sudakov showed against Young Boys why he is expected to be the club's next big-money sale.
Both teams came into the game without scoring a goal in the first three rounds of the new league phase format.
Swiss champions Young Boys are struggling in 10th place in their domestic title defence and took an unexpected lead in Gelsenkirchen, Germany, where Shakhtar are playing their Champions League home games during the military invasion by Russia.
But the Ukrainian outfit recovered to seal the victory.
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