In the day's other Group A encounter, Venezuela were on the wrong end of two controversial decisions in their 1-0 defeat by defending champions Colombia.
Palacios, a stalwart for the last decade in the Peru team, was surprisingly left on the substitutes bench by coach Paulo Autuori but Peru were dismal in his absence, apparently suffering a bout of stage fright.
Bolivia, who rarely offer much of a threat away from the high altitude of La Paz where they play their home matches, went 2-0 ahead in under an hour as they capitalised on Peruvian nerves.
Joaquin Botero put them ahead in the first half on the break before Lorgio Alvarez pounced on a blunder by Peru keeper Oscar Ibanez.
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INSPIRED EFFORT
Palacios replaced a tired-looking Nolberto Solano almost immediately and within six minutes, Bayern Munich striker Claudio Pizarro reduced the arrears from the penalty spot.
Palacios completed the comeback when he flicked the ball up with his right foot and volleyed home with his left with four minutes to play, an inspired effort in the midst of Peruvian desperation.
The match was the second half of a double bill in Lima's National Stadium.
Colombia kicked off the tournament with a fortuitous victory over neighbours after Tressor Moreno slotted home a penalty midway through the first half.
Brazilian referee Marcio Rezende de Freitas awarded the spot kick after Gilberto Angelucci brought down Moreno but television replays suggested the Colombian began his fall before contact was made.
Venezuela dominated the second half and had a goal disallowed for offside when Freitas gave the benefit of the doubt to the defending team.
They also found Colombia goalkeeper Juan Carlos Henao in magnificent form.
The tournament continues on Wednesday as Mexico face Uruguay in Group B action in Chiclayo, while Argentina meet Ecuador later on the same pitch.
Mexico coach Ricardo Lavolpe said he feared the Uruguayans would be even more physical than usual as they attempt to make amends for their dismal recent form.
"I've seen their last games and there's no doubt they're tough," Lavolpe said.
Ominously for Mexico, Uruguay defender Paolo Montero, who has smashed records for red cards and suspensions in Italy's Serie A where he plays for Juventus, will make his international comeback after a two-year absence.
Former Argentina coaches Carlos Bilardo and Cesar Luis Menotti will both be among the crowd as Argentina face Ecuador.
The two great rivals arrived in Chiclayo on the same plane and avoided contact, according to Bilardo, whose ruthless approach to the game contrasts sharply with Menotti's belief in stylish, attacking football.
Brazil, who have rested their top players, open their campaign in Group C against Chile on Thursday in Arequipa and are likely to be missing Manchester United midfielder Kleberson after he strained a thigh in practice.
Kleberson, who played in the knockout stages of the 2002 World Cup, had been hoping to use the tournament to win back his place in the side after a disappointing season in England.
