Born in Tacubaya, Mexico City, Solana came from a family already active in motorsport: his first competitive outings came in a 1954 special known as the Solana Sports, built by Javier Solana. He was also a proficient jai alai player, and earnings from that sport partly financed his early racing career. His motorsport beginnings reportedly included an appearance at the Carrera Panamericana, where a race report cited him as among the most impressive young prospects in the field.
Solana made his Formula One World Championship debut on 27 October 1963 at the Mexican Grand Prix, driving a BRM P57 entered by Scuderia Centro Sud. That race carried a particular historical distinction: Solana started with the car numbered 13, making him the only driver in the history of the Formula One World Championship to start a World Championship race in a car bearing that number — a distinction that stood until Pastor Maldonado adopted 13 as his permanent number under the 2014 regulations. Despite his engine failing eight laps before the finish, Solana was classified as a finisher in 11th place.
Over the following seasons he drove for Team Lotus (1964, 1965, 1967, 1968) and Cooper-Maserati (1966), competing in eight World Championship Grands Prix in total and one non-championship Formula One race. His best championship finishing positions were 10th (at the 1964 Mexican Grand Prix) and 12th (at the 1965 United States Grand Prix). His qualifying record included a 7th on the grid at the 1967 United States Grand Prix. He scored no championship points across the full run.
Team Lotus had sufficient regard for Solana to give him entries in the Lotus 25, Lotus 33, and Lotus 49. The esteem in which he was held by his peers is illustrated by the reported episode in which Jim Clark gifted him a pair of racing gloves as a personal gesture after Solana outpaced him in a free practice session at the 1967 race.
In 1968 Solana also competed in Formula Two with Team Lotus at the Jarama Circuit near Madrid. That same year he was invited to test a Formula Two car for Ferrari — a test that did not result in a signing but demonstrated the level of interest his talent had attracted at the highest levels of the sport.
Away from Formula One, Solana was active in American road racing. He drove for Lola and McLaren in the United States Road Racing Championship and Can-Am series between 1966 and 1968. In March 1968, at the Autódromo Hermanos Rodríguez, he won the first point-scoring race of the USRRC Group 7 series — which was simultaneously the first international sports car race held in Mexico City. Racing Sports Cars records show results in Lola T70 and McLaren M6B machinery across circuits including Riverside, Laguna Seca, and Montjuich.
On 27 July 1969, Solana was killed during the Hillclimb Valle de Bravo-Bosencheve in Mexico. His McLaren ran wide at a bend, struck a concrete trimming at the road's edge, overturned, and caught fire. He was 33 years old.
The first chicane at the Autódromo Hermanos Rodríguez in Mexico City — turns 1 through 3 — is named the Ese Moisés Solana in his honour. The Solana family has remained active in Mexican motorsport and has continued to manufacture handmade sports cars on a largely one-off basis. His career records in Mexican road racing categories and at the Magdalena Mixhuca circuit remained unbroken at the time of his death.