Bergkamp, 36, created all three first-half goals as the second-placed Gunners effectively finished the game off by the 37th minute.
So many Arsenal players were clearly playing for FA Cup final places. The urgency of their performance proved that. And if this was Bergkamp's last game at Highbury, he could not have gone out on a higher note.
Everton were among the favourites for relegation after Arsenal won 4-1 at Goodison in their opening game of the season. Instead, David Moyes' team have earned a Champions League preliminary place, despite losing Wayne Rooney to Manchester United, Tomasz Radzinski to Fulham and Thomas Gravesen to Real Madrid.
Arsene Wenger said Moyes, his staff and players had worked a miracle.
But Bergkamp-inspired Arsenal swept them aside. The Gunners were ahead by the eighth minute when Bergkamp and Dutch youngster Robin Van Persie combined for a picture goal. Bergkamp threaded through a brilliant pass, Van Persie timed his run to perfection and fired a typical left-foot finish past ex-Gunners keeper Richard Wright.
Bergkamp split Everton open for Arsenal's second. Jose Antonio Reyes crossed low. Wright beat out Robert Pires' shot, but the Frenchman looped a clever header over the keeper.
Everton were full of exertion, as usual, but they have lacked penetration all season, especially when Tim Cahill is injured. James Beattie was their only attacker to test Jens Lehmann.
Alessandro Pistone's last-ditch tackle robbed Reyes after Edu's pass climaxed a rapid counter attack. David Weir blocked Bergkamp's shot from the corner.
Wright saved low as Van Persie after a sweeping break by Bergkamp and Vieira.
Vieira blasted wide as Edu's dummy fooled the Everton defence.
Vieira scored the inevitable third. Bergkamp's delicate flick shredded the Everton defence. The Arsenal skipper ran on and dinked his shot over Wright.
The Everton keeper made fine saves from Reyes and Van Persie as half-time approached.
Bergkamp split the Everton defence again in the 48th minute. Sub Thierry Henry went down under Joseph Yobo's challenge. Everton were lucky to escape a penalty.
Two minutes later, Henry nicked a pass off Lee Carsley and Pires nipped in to stroke the fourth.
Bergkamp's vision set up another chance for Henry who just failed to connect. Wright palmed away another Reyes effort, then saved Everton again as Henry bore down. When Henry's shot bounced off Carsley's arm, referee Wiley pointed to the spot. Edu smacked in the kick.
And when Bergkamp charged down Weir's 77th-minute clearance to shoot beyond Wright, the Gunners' fans went wild.
Mathieu Flamini stabbed in the seventh after Reyes' cut back from Henry's thrilling burst. This was Arsenal's biggest-ever Premiership victory.
But the good news for Wenger extended far beyond the scoreline. Henry, out for five games with groin trouble, looked as fresh as ever. Sol Campbell came through successfully after a long-term ankle problem. Edu impressed.
Hip victim Freddie Ljungberg should be fit to face Birmingham on Sunday. So will Gilberto Silva, who missed this game with an ankle knock.