André was born in Alès in the Cévennes, into a mining family. His father was an amateur musician. André studied trumpet with a friend of his father, who suggested that André be sent to the conservatory. In order to gain free admission to the conservatory, he joined a military band. After only six months at the conservatory, he won his first prize.

At the conservatory, André's professor beat him out of frustration and told him to return when he could excel in his playing. A few weeks later, he returned to play all fourteen etudes found in the back of the Arban's book without a single mistake. Maurice André won the Geneva International Music Competition in 1955 and the ARD International Music Competition in Munich in 1963. He was made an honorary member of the Delta chapter of Phi Mu Alpha Sinfonia at Ithaca College in New York in 1970.

André rose to international prominence in the 1960s and 1970s with a large series of recordings of Baroque works on piccolo trumpet for Erato and other labels. Not content to limit himself to the standard Baroque trumpet repertoire, André also performed many transcriptions of works for oboe, flute, and even voice and string instruments. These recordings were (and remain) very popular, and were a strong component of the rebirth of interest in Baroque music in the 1960s. André had over 300 audio recordings to his name, from the mid-1950s to his death.

He had three children: Lionel (1959-1988), trumpeter and music teacher, Nicolas, who plays the trumpet; and Béatrice, who plays the oboe. All three performed with their father in concert. He also made several recordings with his brother Raymond (b. 1941).

One of André's students, Guy Touvron wrote a biography on him entitled Maurice André: Une trompette pour la renommée (Maurice André: A Trumpet for Fame), which was published in 2003.