The Gunners were in control of the game for the most part in North London, although Celtic played right to the end and nearly came back from a seemingly impossible position.
Arsenal came firing out of the blocks and took the lead in the third minute when Carlos Vela poked home at the far post after Theo Walcott fired the ball across goal.
And Sagna made it two on the stroke of half-time when his left-footed shot from the right edge of the area beat Hoops goalkeeper Lukasz Zaluska at his near post.
Nasri slotted home to make it three after the break, as Georgios Samaras then missed a penalty before Daryl Murphy and Ki Sung-Yong set up an exciting finish.
The result means Arsene Wenger's side finish top of the table, winning the pre-season tournament ahead of Lyon, AC Milan and Celtic.
The hosts started the game well and Walcott gave Celtic a sign of things to come inside the first minute when he broke forward to lift the ball to the back post, but no team-mate anticipated the cross.
Walcott was involved in the opening goal, as the impressive Jack Wilshere slid him into the area before the England international fired the ball across goal for Vela to poke into an empty net.
The goal lifted Arsenal and the Gunners were in complete control from thereon in, and Hoops keeper Zaluska had to be alert to palm away a Tomas Rosicky free-kick.
Walcott was causing the Celtic defence all sorts of problems but it was defender Thomas Vermaelen who had the next effort - lashing a shot on target from long range that was saved by Zaluska.
Neil Lennon's men did venture forward in the 17th minute and Daryl Murphy found himself in a good position inside the area after a neat run, but just when composure was needed he sliced the ball high and wide when off balance.
New signing Efrain Jaurez looked dangerous whenever he broke into attack, and he had a shot wide before Manuel Almunia punched a deflected Charlie Mulgrew free-kick clear.
Walcott then stung the hands of Zaluska with a free-kick of his own and the Celtic keeper produced a fine save in the 34th minute to tip over a deflected Wilshere effort from inside the box.
From the resulting corner, Wilshere was denied an absolute blinder as he hit a first-time volley that smashed the crossbar, with Zaluska possibly getting a crucial touch.
But the Polish shot-stopper was at fault just a few minutes later as Sagna struck a left-footed shot from the edge of the area to double Arsenal's lead.
The French defender's effort was hit well but Zaluska was slow to get down at his near post, although he may have been unsighted.
Arsene Wenger brought on Nasri and Marouane Chamakh at the break and it only took Nasri five minutes to make a decisive impact.
Gael Clichy won the ball well and Nasri played it out wide to Walcott, who cut inside and slipped it back to the midfielder who slotted home with a cutting low finish.
Substitute goalkeeper Dominic Cervi had only been on a few minutes but already he was picking the ball out of his own net.
Celtic responded well and showed character after going three down, and had a golden opportunity to pull one back from the penalty spot after Wilshere dragged down Marc-Antoine Fortune inside the area.
However, Greek striker Samaras blazed the spot-kick over the bar in front of his own supporters.
The travelling fans did have something to cheer in the 75th minute, though, as Murphy grabbed a consolation goal from close range.
Ki struck a stinging drive and it deflected into the path of Murphy, who reacted quicker than a static Arsenal defence to fire past Almunia.
Celtic then set up a grandstand finish as Ki finished with aplomb after Fortune broke down the right and fed him in with a pinpoint low cross.
But they could not get the crucial equaliser and despite saves from Almunia, Arsenal held on for the victory.