Cavani ran onto Walter Gargano's weighted pass and beat two defenders before unleashing an unstoppable left-footer into the bottom corner at the Estadio Cachamay here Tuesday.
Venezuela were reduced to 10 men with five minutes remaining when Tomas Rincon received his second yellow card for a rough challenge on Cristian Rodriguez.
The result lifts Oscar Tabarez's side to fifth in the South America zone standings with 16 points from 12 matches.
The top four teams earn direct qualification for the World Cup in Brazil while fifth will play off for a spot against the fifth best side from Asia.
In other matches Tuesday Chile defeated Bolivia 3-1, Colombia beat Peru 2-0 while Ecuador drew 1-1 with Argentina.