The Gunners, second in the table on goal difference, strolled to the win over Joe Royle's men, who failed to seriously test third choice goalkeeper John Lukic all afternoon in a poor display.
The game turned in the space of a minute shortly before the break. Danny Tiatto, who had earlier been booked for a tackle on Oleg Luzhny, took down the flying Thierry Henry 25 yards out was shown a second yellow card and sent off.
The hammer blow came seconds later when, from the free kick, impressive youngster Cole fired a low shot past Nicky Weaver and into the bottom corner.
Bergkamp added a second on 52 minutes when he curled a magnificent shot into the bottom corner before substitute Sylvain Wiltord beat Weaver to add a third, with Henry scoring twice late on to add a gloss to the win.
New City signing Laurent Charvet joined Spencer Prior and Steve Howey in defence, but the former Newcastle man seemed unable to stop relentless Arsenal pressure from turning into goals.
The first half was a poor one, with City unable to create any real chances and Arsenal dominating a midfield battle, before the hammer blow of Tiatto's sending-off changed the game.
Henry was a constant danger but could not find the target with a number of efforts, and Prior was relieved to see the lively Robert Pires substituted after 21 minutes.
Ray Parlour, whose running was a constant danger, and Gilles Grimandi both had efforts off target as the period drew to a close.
At the other end, Shaun Goater, starting his first Premiership game, forced Lukic into a diving save from 25 yards, but City could not find any space inside the area, missing the maverick of Paulo Wanchope.
Bergkamp, who had an unusually quiet game, scored the second within seven minutes of the restart, and the game was effectively over as a contest.
Patrick Vieira forced Weaver into another impressive save with a left-footed effort from outside the area before Henry fired over the bar.
City's best chance of the game saw Goater inexplicably miss an open goal from three yards after 63 minutes when Paul Dickov squared the ball after Jeff Whitley's sliced effort fell to him, the Bermudan lifting the ball over the bar with the whole goal at his mercy.
Substitute Mark Kennedy was unable to create the space for a shot after twice getting himself into good positions, but in truth City were poor on attack all afternoon, lacking presence against the experience of returning duo Tony Adams and Martin Keown.
The introduction of Wiltord with a quarter of an hour remaining ensured that Arsenal kept the pressure on, and his pace earned him a goal within minutes of coming on, Henry playing him through before he coolly finished through Weaver's legs.
The England Under-21 international made a series of good saves for City, but was powerless to prevent Henry grabbing a deserved brace late on.
For his first, he controlled Wiltord's cross with his back to goal and turned to fire home a half-volley from five yards with some style.
He then broke from his own half before playing a one-two with Wiltord, lifting the ball over the onrushing goalkeeper as the game drew to a close.
There was still time for Ljungberg and Henry to go close but City return north knowing that much improvement is needed if they are to retain their Premiership status, while Arsenal continue to press their title claims at the other end of the table.
