Gaston Sirino steered the Citizens to a 1-0 first leg win with his goal at the Moses Mabhida Stadium last week.
Kyle Jurgens added to the scoreline with a headed goal at the Stade Intwari on Sunday inside the first minute.
Samkelo Maseko then fired home in the 37th minute to hand Khalil Ben Youssef's side an unassailable lead.
The Burundi Premier League side did manage to reduce the deficit, but the Betway Premiership side were ultimately deserved winners.
City will now face Mauritian side Port Louis in the next round, after they defeated Botswana’s Jwaneng Galaxy 2-1 on aggregate.
