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.
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.
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.
