Siva Sakthi (10th minute) and Brazilian Alan Costa (61st) scored for the winners, while Apuia scored the lone goal for the Mumbai side in an entertaining match.
The Blues took an early lead with a fifth tournament goal from Siva Sakthi.
CHAMPIONS, AGAIN! 🏆 Bengaluru, your Blues bring home the #DurandCup. 🔥 #WeAreBFC #MCFCBFC https://t.co/aHaOvjvooM
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The long ball was delivered from the deep end by Jovanovic, putting pressure on Mourtada Fall with Siva on his shoulder. The Senegalese defender was eventually foxed by the bounce and Siva jumped it with a measured chip over the keeper.
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After Chhangte had the first chance for the Islanders in the 17th minute, they equalized on the half hour mark when Sandesh Jhingan conceded a free kick in a dangerous zone.
Ahmed Jahouh played off Greg Stewart, who went on one of his runs and hit a left footed grounder towards goal, which fended off Gurpreet Sandhu. However, Apuia was on hand and tapped in the rebound with ease.
The Blues then almost took the lead in the 38th minute, when Roy Krishna rebounded from a corner from Roshan Singh and shot on target, but Chhangte made a wonderful save on the goal line.
Vinit Rai then got a chance after being outplayed by Stewart but shot over from close range and the only other notable incident of the half was the first yellow card of the game for Mumbai captain Mourtada Fall who had a memorable 45 minutes .
The game remained tense in the second half. Five minutes into the restart, Jahouh delivered a measured long ball that fell for Chhangte, who did well to get away from two BFC defenders, but with only the keeper to beat, the highest scorer of the tournament went over.
BFC then took the lead for the second and final time in the game when Brazilian Alan Costa jumped highest from a corner of Sunil Chhetri aimed at the far post and nailed a header into the back of the Mumbai net.
The most notable opportunity for Mumbai in the second half came via Spanish midfielder Ripoll in the 77th minute as he ran solo from midfield and unleashed a thumping drive that defeated Gurpreet but knocked the left to his feet and bounced back.
Chhetri also had a couple of golden opportunities to score, once in the 69th minute when his left foot missed the goal and then again in the 87th when he was one-on-one with the keeper, but Lachenpa was ready and brought a great rescue.
With six minutes of extra time, Mumbai gave one last chance and Greg Stewart came close in the 94th minute, but his effort also went just wide.