Benfica went into the game knowing even victory may not be enough to send it into the knockout stage.
Both goalkeepers were called into action in the energetic opening minutes.
Enzo Perez gave Salvatore Sirigu an early test with a curling shot that had the PSG goalkeeper leaping high to his left. Then Silvio forced Sirigu to dive sharply low to his left as Benfica pressed forward. Cavani set up Jeremy Menez for a low shot from the left that Artur was at full stretch to keep out.
The game cooled as PSG held onto possession, though Gaitan's shot just beyond Sirigu's post in the 27th looked to be going in.
Benfica's push forward occasionally left it exposed at the back, and Ljubomir Fejsa made a last-ditch tackle to deny Lucas Digne inside the home team's penalty area.
Benfica wasn't giving up, and Gaitan's winner capped a period of renewed pressure as the home team repeatedly pierced the visitor's defense.
PSG still threatened, though, and Romuald Lacazette's shot scraped the post in the 68th.