Arsene Wenger's men exited the Champions League at the last 16 stage to Monaco on away goals despite a 2-0 victory at the Stade Louis II on Tuesday.
But they quickly rebounded from that disappointment by overcoming a depleted Newcastle side on Saturday, as Giroud made it eight goals in six games against the Tyneside club.
The France striker scored his brace in the space of four first-half minutes, breaking the deadlock with a cleverly flicked effort into the bottom-right corner.
Giroud then made it 2-0 by capitalising on some slack Newcastle defending to head home and give the hosts a mountain to climb.
Moussa Sissoko provided some hope for the St James' Park crowd when he halved the deficit shortly after the half-time interval.
Yet, despite a much improved second-half showing, Newcastle could not complete the comeback as Arsenal stayed within a point of second-placed Manchester City.
Arsenal had slightly the better of an open start to the contest but Newcastle's makeshift backline, which saw Daryl Janmaat move to centre-back and midfielder Jack Colback take his right-back spot, coped well with the early pressure until Giroud pounced in the 24th minute.
Giroud demonstrated good striker's instincts to deflect Danny Welbeck's header from Santi Cazorla's right-wing free-kick beyond Newcastle goalkeeper Tim Krul.
And, shortly after Welbeck had volleyed an Alexis Sanchez cross wide, Giroud doubled his tally.
Again Newcastle failed to deal with a set-piece, with Giroud seeing off the attention of defender Mike Williamson to nod Cazorla's delivery into the bottom-left corner.
Newcastle's effort did not drop after falling two behind and they were rewarded three minutes into the second half by Sissoko, who met Remy Cabella's right-wing pull back with a cool first-time finish.
Ayoze Perez then curled over the crossbar with a spectacular 25-yard strike before Yoan Gouffran shot straight at David Ospina from close range.
Ospina was called into action again, producing an excellent reaction save to keep out Mike Williamson's header, with Janmaat the next to go close with a vicious long-range drive.
Newcastle continued to push forward in search of a leveller, but Arsenal stood firm to put the pressure on fourth and fifth-placed Manchester United and Liverpool, who meet at Anfield on Sunday.
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