Four times finalists Argentina will meet world number one Novak Djokovic’s Serbia, the 2010 champions, in another home tie in the last eight on July 17-19.
Argentina hope to have former U.S. Open champion Juan Martin del Potro back in the team if he has recovered from a wrist injury.
“Leo's win yesterday (Sunday) allowed me to play today for the victory. Thanks to the crowd for their support,” Delbonis said courtside.
Argentina had flirted with defeat by their bitter South American rivals when they were 2-1 down after the doubles and number one Leonardo Mayer needed 11 match points to overcome Joao Souza in the fourth rubber which lasted a competition singles record six hours and 42 minutes on Sunday.
Delbonis reached his first match point after a Bellucci drive landed over the baseline after one of the many long rallies in a tense final rubber which began on Sunday.
He sealed the win with a fine drive down the line out of the Brazilian's reach.
The Argentine number two carried a first set lead into Monday’s extra day on the purpose built claycourt at the Tecnopolis exhibition centre on the city outskirts.
Delbonis lost the second set but his greater accuracy with drives and backhands from the baseline and Bellucci’s higher number of unforced errors, especially when at the net, saw him through to victory.
With a colourful group of Brazilian fans no longer in Buenos Aires on Monday, the court was packed full of Argentine support that unsettled Bellucci, loudly cheered every Delbonis point and launching into soccer style chants at every interval.
Brazil, who reached the World Group when they beat Spain in last year's promotion-relegation round, will looking to retain their status in the same playoffs later this year.
(Reporting by Rex Gowar, editing by Ed Osmond and Justin Palmer)
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