Messi, who helped Miami win the 2023 crown in his first weeks with the Florida club, took his tally in the competition for Major League Soccer and Mexican Liga MX sides to a record 14 goals.
David Rodriguez gave San Luis the lead in the fourth minute but Messi equalized seven minutes later.
Venezuelan midfielder Telasco Segovia edged Miami ahead in the 26th minute.
Messi struck again just before half-time with Brazilian defender Micael making it 4-1 deep into first-half stoppage time.
Atletico's Rafa Llorente scored a second-half consolation.
Other Leagues Cup matches on Wednesday saw Dallas beat Queretaro 2-0 and Leon won 1-0 at Nashville.