The already-relegated Cottagers came brought Tom Cairney back into the team, while there was also a single alteration for the visitors who included Leandro Bacuna.
Neither side started with any pace, Fulham trying to build momentum but lacking quality, while Cardiff were happy to sit deep and soak up pressure.
Aleksandar Mitrovic had the best chance of the first half when he forced a good save from Neil Etheridge on the turn but the main talking point from the first half was an injury to Denis Odoi.
The Belgian was tracking back towards his own goal and collided with team-mate Maxime Le Marchand. Treatment was required and resulted in nine minutes of stoppage time before the break.
Things continued in the same vein in the second half but the breakthrough came on 79 minutes when Cyrus Christie worked down the right before rolling back to Babel who displayed quality unfamiliar with the rest of the game and whipped home past Etheridge from the 18-yard line.
That seemed to wake Cardiff up and they had a string of chances, the best seeing Junior Hoilett striking the underside of the bar with Sergio Rico beaten.
Rico earned his wages elsewhere though, saving twice from Sean Morrison and Danny Ward, while Rhys Healey also had an effort blocked.
With seconds to go, Morrison was again thwarted by the goalkeeper, summing up an afternoon which seemed like a wasted opportunity in their battle for survival.
They are now likely to need to beat Crystal Palace and get something from Manchester United on the final day of the season.