It was an indication of what was to come from the hosts as they controlled the tempo of the game, but a good opportunity was missed when Ljajic fired an effort into the side-netting after earning himself space in the box.
Genoa rarely caused problems for the Inter defence at the opposite end, and the visitors were left thanking Mattia Perin before the break as the Italy international produced a stunning save to keep out a powerful low strike from Ljajic.
The save proved to be crucial as Inter took the lead thanks to Ljajic, whose free kick from the wing evaded the Genoa defence on the way to finding the back of the net.
Genoa battled impressively to stay in the game, but they wasted a good opening on the break when Goran Pandev's poor touch prevented him from getting a shot away against his former side.
As Inter struggled to kill the game off in the closing stages, their task was made more difficult when D'Ambrosio picked up a second yellow card with four minutes of normal time remaining.
However, the hosts held off the rare spells of pressure from Genoa to ensure the pressure was put on Napoli, who visit Bologna on Sunday.