MLS roundup: Messi-less Miami beat Chicago to stretch lead, Red Bulls & Union win
Inter's 16th win in 25 games took them to 53 points, five clear of second-placed FC Cincinnati who lost 3-1 at New York Red Bulls.
Paraguayan midfielder Matias Rojas gave Miami the lead in the sixth minute.
Saudi-born German defender Rafael Czichos equalized for the Fire in the 73rd minute before former Spain star Alba scored the winner two minutes later.
In New York, Cincinnati were stunned by two goals from Scottish midfielder Lewis Morgan either side of defender Kyle Duncan's strike as the Red Bulls led 3-0 after 59 minutes.
Corey Baird got the Cincinnati consolation in the 67th minute.
A hat-trick by Philadelphia's Daniel Gazdag gave the Union a 3-0 win over visiting Nashville.
Gazdag scored in the 10th, 39th and 89th minutes to keep Philadephia one point behind Atlanta for the last East playoff spot.
Norwegian defender Stian Gregersen scored in the 60th and 77th minutes as Atlanta United came from behind at home to beat Columbus 2-1.
Canadian midfielder Richie Laryea scored in the 38th minute to give Toronto a 1-0 victory at Montreal.