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Inst led 6-3 at the break thanks to two Hume penalties while John Crowther slotted over a Campbell penalty.
Campbell piled on the pressure in the second half but could not find a way through a resolute RBAI defence, even when Mark Keane was yellow-carded.
Hume sprinted clear from deep in his own half to run in a try, which he converted in the game's final action.
Campbell will rue missed opportunities during the St Patrick's Day decider in a sun-kissed Belfast.
They dominated the latter stages of the first half and entire second period.
Penalty opportunities were turned down in favour of going for a try but they were thwarted by handling errors and Inst defending.
Campbell launched a final attack but Hume collected from a bouncing ball and raced away to seal a second straight Schools' Cup triumph.
RBAI captain Conor Field said: "I can't describe my feelings after that but I am so proud of the boys.
"Credit must go to Campbell as they put us under so much pressure although we defended strongly throughout the game.
"To see James run clear to win it for us so late in the match was absolutely brilliant."
James Hume scored a late try as RBAI beat Campbell College 13-3 at Kingspan Stadium to retain the Schools' Cup.