Benito Armando Archundia, Mexican referee, who was caught up in controversy after his dubious handling of the United States v Canada Gold Cup semi-final last week, angered Peru by not awarding a second-half penalty.
Fresh from their 3-0 win over Uruguay on the opening day, Peru began well but the game changed in the 14th minute.
Pedro Garcia was involved in a fierce tussle with Venezuela's Giancarlo Maldonado, but the Peruvian was sent off for elbowing while Maldonado, who appeared to start the incident, received a yellow card.
Television replays showed contact was made on the chin, but Maldonado fell over backwards and lay on the ground clutching his face in agony before returning to his feet as soon as Garcia was dismissed.
"Venezuela did very little, we created five or six clear scoring chances in the second half," Julio Cesar Uribe, Peru coach, said.
"The sending-off influenced (the outcome) but we have to accept it as part of football."
The visitors held out until halftime, but four minutes after the break Cichero out-jumped the defence to head in from a corner.
Penalty call turned down
Peru were protesting again in the 62nd minute after Paulo Guerrero won a loose ball and crossed to Pizarro, who appeared to be tripped by Hector Gonzalez as he tried to tap the ball in.
Referee Archunda, who last week infuriated Canada by disallowing what would have been a stoppage-time equaliser in their 2-1 Gold Cup defeat to the US, waved play on.
Richard Paez, Venezuela midfielder, was sent off in the 78th minute, picking up a second yellow card for petulantly kicking the ball away after an offside decision, but a minute later, substitute Daniel Arismendi burst down the left, cut inside and scored with a low shot which crept inside the post.
The match, which saw the hosts secure their first Copa victory in three decades, ended in ugly scenes after Venezuelan defender Alejandro Cichero played keepy-uppy near his own penalty area and was booked, apparently for ungentlemanly conduct.
Cichero's behaviour infuriated Peru, leading to a scuffle on the edge of the area and pushing and shoving as the players left the pitch.
Uruguay win tough
Earlier, Vicente Sanchez's 58th minute goal was enough to give Uruguay the points in the first match against Bolivia.
Uruguay were boosted by Vicente Sanchez,
top, who scored their winning goal [AFP]
However, Bolivia were also upset as US referee Baldomero Toledo failed to punish some wild Uruguay challenges, especially a scything tackle from behind by Carlos Diogo on Jaime Moreno.
Toledo appeared uncomfortable in the pressure-cooker atmosphere of South American football and struggled to control the game.
The results see Venezuela go to the top of Group A with four points, followed by Peru and Uruguay with three apiece and Bolivia with one.