Sergio Aguero, the son-in-law of Diego Maradona, came on as a second-half substitute and promptly rescued a point after Brazil-born Edvaldo Rojas had stunned the hosts by breaking the deadlock just two minutes into the second half. The 25-year-old ghosted in to nudge home following a corner on the left, the ball running under Ever Banega's foot on the line with keeper Sergio Romero beaten.
With Lionel Messi's Barcelona form stubbornly elusive - not for the first time in Albiceleste colours - Argentina struggled to respond before a spell of pressure finally told as Aguero, on just minutes earlier for Ezequiel Lavezzi, superbly blasted them level on 76 minutes following a Nicolas Burdisso chest down.
The opening 45 minutes had seen Argentina enjoy most of the possession and an early succession of corners, which came to nothing, while at the other end a Marcelo Martins free-kick ended up in Romero's midriff.
Despite having on paper a highly potent attack in Messi and Carlos Tevez, the hosts looked short on ideas to start with. The Manchester City man was also short on fuse, earning a booking for dissent on 36 minutes.
Happy to be level at the break, Bolivia came out with a spring in their step and after Rojas put them a goal to the good, Martins should have made it two after 58 minutes. The striker was clean through on goal but showed too much of the ball to a grateful Romero, leaving coach Gustavo Quinteros fuming on the touchline.
Angel di Maria then drew gasps with a low drive flashing just wide as an equaliser loomed ever larger, before Aguero finally delivered.
Argentina will now have to up their game considerably when they meet Colombia in Santa Fe on Wednesday, before their final group match against special guests Costa Rica.
Sergio Batista's men came into the tournament under pressure to deliver the goods after their humiliating 2010 FIFA World Cup South Africa™ defeat at the hands of Germany - a loss which saw Batista's former Mexico 1986-winning team-mate Diego Maradona dispensed with as coach.
Bolivia belie the stats
In their Copa America curtain-raiser, Argentina were also mindful of an even worse, if somewhat less public, humiliation than the one Bolivia inflicted upon them in their last meeting, when the underdogs emerged 6-1 winners in a FIFA World Cup qualifier in La Paz in 2009. Even so, the odds were stacked against the visitors, whose only title came on home soil in 1963.
Each of 11 previous face-offs in Buenos Aires went the way of the two-time world champions, with a goal count of 35 for only three conceded, but this time they had to make do with stalemate.
Although Brazil, who start their campaign in La Plata against Venezuela on Sunday, have won only eight continental crowns to 14 for Argentina and Uruguay, they have bagged four of the last five editions, with Colombia interrupting the sequence in 2001.