Kabba, on loan from Championship rivals Watford, opened his account for the Seasiders with a stunning strike from outside the box with just four minutes remaining.
Both sides have been tipped as relegation fodder but managed to produce a decent contest in a game neither team would really have wanted to lose.
The home side dominated throughout and had the clearer chances, most notably having two close-range efforts headed off the line in the second half.
The visitors, who were marshalled by debutant Hugo Colace, looked to be extending an unbeaten run against the Seasiders going back 11 games.
Simon Davey's side did have one good first-half chance but Jon Macken headed over from eight yards.
Blackpool had an even better first-half chance but saw leading scorer Ben Burgess blaze over from a similar distance.
The visitors did think they had earned a penalty for a handball midway inside the second half but, once referee Tony Bates waved away the appeals, it was the home side who turned up the heat.
Firstly, Stephen Foster somehow headed off the line to deny sub Keith Southern.
Then Robert Kozluk was on hand to deny ex-Tykes striker Daniel Nardiello with another stunning goalline header with just five minutes remaining.
Nardiello was denied once again against his former club with a brilliant near-post save by Luke Steele.
Then, just as the Tykes' never-say-side defending looked to have earned a point, up popped Kabba to snatch a priceless win.