France striker Giroud, who had not scored in all competitions since Nov. 23, got the faintest of touches to glance a Theo Walcott free kick into the net midway through the second half.
Romelu Lukaku, who had not scored for Everton in five games, scored the winner for Roberto Martinez's side three minutes after Gaston Ramirez cancelled out Seamus Coleman's early opener.
Arsenal, with 42 points, are one point clear of Manchester City after 19 games. Everton, in fourth on 37, leapfrog rivals Liverpool, who play Chelsea later on Sunday.
Tottenham Hotspur host Stoke City in Sunday's other match.
Arsenal's visit to Newcastle promised goals after the North Londoners won 7-3 when the teams clashed on the same day last year, but this encounter soon descended into a more stolid affair with both defences well on top.
Arsenal kept possession neatly, but without injured playmaker Mesut Ozil, lacked incision and failed to create a chance of note in the first half.
Newcastle came close to taking the lead in first-half injury time when Arsenal keeper Wojciech Szczesny palmed a shot from Moussa Sissoko over the bar and defender Mike Williamson hit the crossbar with a header from the resulting corner.
It looked to be heading towards a draw when Arsenal struck the decisive blow. Walcott's free kick was perfectly angled into the box and Giroud got in between the Newcastle centre backs to feather a delicate header into the net.
It could have been 2-0 moments later but Walcott missed two chances to put the game to bed, with Newcastle keeper Tim Krul saving his first shot while his follow-up effort was cleared off the line by Mathieu Debuchy.
There was a frenetic finale but Arsenal's defence held out to ensure they go into the New Year on top of the table for the first time since 2007-08, to provide manager Arsene Wenger with some festive cheer.
"There was a big satisfaction because we surprised many people and we want to go on from here," he said.
"We played two away games and got six points against West Ham and Newcastle and as you can see the team is ready for a fight. Let's keep fighting and trying to develop the team."
Everton, who played more than 60 minutes of their defeat to Sunderland on Thursday with 10 men, seemed to be rejuvenated with their full quota on the field and took the lead after nine minutes.
Seamus Coleman, cut inside, switched the ball to his right foot and fired an angled shot into the net for his fourth goal in his last seven Premier League games.
Ross Barkley missed an easy chance to make it 2-0 early in the second half from eight metres and that appeared costly when Ramirez levelled with a thundering shot from distance.
Parity lasted only a few minutes as Lukaku ran onto a pass from James McCarthy and the Belgium striker took one touch and fired home from 12 metres.
