Boca, with Daniele De Rossi watching from the bench, opened the scoring in the 11th minute, Ramon Abila getting on the end of an Alexis Mac Allister pass to make it 1-0.
LDU Quito were dealt a huge blow just before half-time as Jefferson Orejuela was sent off for a rash challenge.
Emanuel Reynoso doubled Boca's lead with a brilliant 25-yard free-kick, and Kevin Minda was sent off while on the bench for the hosts.
Boca sealed a fine win with nine minutes remaining, Luis Caicedo's poor defending eventually leading to the defender scoring an own goal.
If Boca and rivals River Plate, who play Cerro Porteno on Thursday, advance, they will meet in the semi-finals in a rematch of last year's controversial decider.
In the day's other game, Bruno Henrique's late brace saw Flamengo beat Internacional 2-0 at the Maracana.
Henrique tapped in a Gerson pass for the 75th-minute opener before producing a fine finish soon after to seal Flamengo's first-leg victory.