Victory took them back to second in the table. The champions are ten points behind leaders Chelsea and one ahead of Manchester United, who visit Highbury on February 1.
Arsenal should have wrapped up three points long before the end. A combination of missed chances and Shay Given's brilliant goalkeeping kept the home fans on the edge of their seats.
The Gunners were almost made to pay in an 81st minute breakaway, but Lee Bowyer fired Newcastle's best chance straight at Manuel Almunia.
Arsene Wenger made four changes from the team that lost at Bolton last weekend.
Lauren returned from a knee injury. Bergkamp and Jose Antonio Reyes, subs at Bolton, both started. Mathieu Flamini lined up in midfield instead of Cesc Fabregas. Freddie Ljungberg was out with a hamstring strain.
Newcastle were without Craig Bellamy, apparently because of a hamstring injury sustained on the training ground. Steve Carr and Nicky Butt remain long-term casualties.
Arsenal almost scored inside 30 seconds when Flamini fired Robert Pires' header against Given from close range. Reyes and Thierry Henry tested the Newcastle defence before Bergkamp made the breakthrough. Flamini collected Kolo Toure's long pass. His clever flick found Bergkamp who controlled and shot past Given in one movement.
Given saved Newcastle again two minutes later with a flying save to palm away Henry's 20-yarder. Bergkamp, shooting instantly, was wide of the near post and Pires curled a 35-yarder inches wide with Given struggling.
Bergkamp then just failed to connect with Pires' cross, before Lee Bowyer and Shola Ameobi were both off target in rare Newcastle attacks.
Arsenal stepped up a gear in the second half. Given made a fine save from Henry's free-kick as Flamini blocked his view, then Toure fired wide from Bergkamp's pass following the rebound.
Given saved at Reyes' feet after another precise pass by Bergkamp. His best save came in the 61st minute after the outstanding Patrick Vieira split the Newcastle defence. Given's touch deflected Henry's shot against an upright and the ball bounced back into the keeper's hands.
He made a flying stop from Pires' stab after a sweeping break by Reyes and Vieira, then foiled Henry again.
Toure and Vieira flashed headers wide from corners, before Given completed Henry's frustration with a 75th minute save after the Frenchman fired a seemingly unstoppable shot on the turn.
Arsenal have been pegged back too often this season in games they have controlled. Not this time.
Manager Graeme Souness has instilled a harder edge to Newcastle's play. Some of their first-half tackling had Wenger jumping from the dug-out.
Alan Shearer needs one goal to become the first player to score 250 in the Premiership. He had only one chance, a stoppage time header that flew wide of target.