De Jong hit the only goal of the match in the 36th minute while Hannover's night was further soured when Sergio Pinto was sent off for a second booking in the 85th minute.
The win sees the hosts extend their unbeaten run to four games to move onto 38 points, one away from a Europa League berth.
Gladbach's first chance arrived after 20 minutes but De Jong could not beat Ron-Robert Zieler.
De Jong did get the job done in the 36th minute, though, Patrick Herrmann playing a perfectly-timed chipped through-ball for the Dutchman to collect and poke under the onrushing Zieler.
Hannover nearly equalised in the 45th minute but Gladbach goalkeeper Marc-Andre ter Stegen was able to get his hand on Sebastian Pocognoli's powerful effort from 35 yards to deflect the ball onto the crossbar.
Gladbach had an opportunity to extend their lead 10 minutes after the re-start as Amin Younes fed Herrmann but he shot wide of the left upright.
Hannover were struggling to create chances although Didier Ya Konan worked a good position for a header in the 73rd minute but he steered his effort wide.
Hannover went a man down with five minutes left as Pinto picked up a second booking, but they nevertheless were able to carve out one more chance to salvage a point as Christian Pander took aim with a powerful effort but Ter Stegen saved well.