Quilmes went two goals up in the opening eight minutes, stunning coach Carlos Bianchi on his return to La Bombonera and a crowd including playmaker Juan Roman Riquelme, who was welcomed back on Friday from a seven-month break.
Defender Wilfredo Olivera headed the opener in the fifth minute from a set piece and striker Cristian Menendez added the second in a counter-attack.
Midfielder Walter Erviti got Boca back into the game eight minutes before halftime and Burdisso equalised in the 67th.
Boca had Matias Caruzzo, Burdisso's partner in central defence, sent off in the 78th minute after bringing down Martin Cauteruccio but Orion saved the Uruguayan striker's penalty.
Three minutes later Burdisso scored the winner when he met a corner on the edge of the box and volleyed it superbly into the net.
"Those were the first two goals I've ever scored with my foot," Burdisso, noted for his heading power, told reporters.
The floodlights failed at both matches on Friday though that did not stop title holders Velez Sarsfield from getting off to a winning start.
Velez, winners of the 'Inicial' championship in December, beat Argentinos Juniors 1-0 at the Diego Armando Maradona stadium with a fine strike from 30 metres by Lucas Pratto.
Argentinos had started strongly but a power failure after 12 minutes caused a delay of more than half an hour and on their return, Velez had tweaked their tactical plans and took control.
The bad news for Velez, who are eyeing a second South American Libertadores Cup this year, was the loss with a serious ankle injury of Pratto's strike partner Facundo Ferreyra, joint top scorer of the 'Inicial' with 13 goals.
The lights also failed causing a 20-minute delay in the second half at the '15 de abril' stadium in Santa Fe where Union and Arsenal drew 1-1.
Arsenal winger Pablo Luguercio earned the first red card of the championship for slapping Emanuel Britez after the Union defender had touched his buttocks as he was shielding the ball.
Boca's arch-rivals River Plate visit Belgrano in Cordoba on Sunday.