Venegas was born in a poor section of San Juan, Puerto Rico where fighting was a common way of life. Many of the youth at that time saw the sport of boxing as a way to a better life. Among them was Venegas, who admired Puerto Rico's first international boxing champion, Sixto Escobar. Venegas took up boxing and his boxing abilities soon called the attention of the islands young Olympic Committee. In 1948 the Puerto Rican Olympic Committee included him in the delegation which would represent the island.
The 1948 Summer Olympics celebrated in London, was a historical one for Puerto Rico because it was the first time that the island would participate as a nation in an international sporting event. The island's delegation consisted of twelve members. It should be noted that in their opening ceremonies, the Puerto Rican delegation carried the flag of the United States into the Olympic stadium. The United States protested, claiming that two nations could not use the same flag at the same time. The decree of Commonwealth on July 25, 1952 would give the Puerto Rican delegation a flag of their own.
In the 1948 Summer Olympics, known as the XIV Olympics, Juan Evangelista Venegas made Puerto Rican sports history by winning Puerto Rico's first Olympic medal ever when he beat Belgium's representative, Callenboat, on points for a unanimous decision. He won the bronze medal in boxing in the Bantamweight division, falling short of the silver medal to Giovanni Zuddas. 
After returning back to a hero's welcome to Puerto Rico, Venegas turned professional. In 1949, Venegas made his professional boxing debut, but his boxing career was a brief and short one. Venegas, a southpaw, fought in the flyweight division for a total of 8 fights, 6 of which he won and 2 that he lost. None of his wins were by way of K.O.'s (Knockouts).
In honor of his memory, The Puerto Rican Boxing Commission sponsors the Juan Evangelista Venegas boxing tournament, which serves as a tune-up to future Olympic boxing propects.