The Brazilian Copa Libertadores champions took a 1-0 lead into the second leg at their Estadio Governador Magalhaes Pinto home and eventually claimed the trophy 5-3 on aggregate.
Lanus levelled the tie in the 94th minute through Lautaro Acosta, but the Argentines scored two own goals in extra time to gift Mineiro the crown.
Acosta was also sent off late in the encounter for the Copa Sudamericana winners.
Mineiro looked on track for a far more comfortable win when Diego Tardelli gave them the lead in the second leg.
Tardelli sent Lanus goalkeeper Agustin Marchesin the wrong way from the penalty spot in the seventh minute.
Victor Ayala drew the visitors level on the night just two minutes later, completing a fine team move with a powerful low finish.
In the 26th minute, Lanus were level on aggregate as Santiago Silva scrambled in after a set piece went uncleared.
Mineiro attacker Maicosuel put away a Marcos Rocha cross from the right seven minutes before half-time to make it 2-2 in the second leg.
The hosts looked like they had hung on for the win, only for Acosta to scrappily send the game into extra-time.
Mineiro took a lucky lead in the 103rd minute.
Substitute Luan, who replaced Ronaldinho in the second half, saw his cross from the left deflect off Gustavo Gomez and past Marchesin.
It got worse for Lanus nine minutes later.
Ayala was under no pressure in his own penalty area when, attempting to header back to Marchesin, his effort went over the goalkeeper and in, sealing Mineiro's win before Acosta saw red.