At a ground where they had only won once on eight previous Premier League visits, the Gunners grabbed the lead three minutes prior to the break through Giroud's close-range strike, with Mesut Ozil then extending their advantage in the 55th minute with a neat finish.
Stoke hit back with a controversial 67th-minute goal from substitute Peter Crouch, before efforts from Alexis Sanchez in the 76th minute and Giroud 10 minutes from time wrapped up a victory that sees the gap between Arsene Wenger's fifth-placed men and fourth-placed Liverpool reduced to a point.
Both sides have two games left, and Arsenal will be watching with great interest when Liverpool play at West Ham on Sunday, hoping their fellow Londoners can do them a favour
Manchester City, also with two matches to go, are three points clear of Wenger's team in third.
There have been various protests of late from Arsenal fans calling for the manager to leave, and early on in this game a plane flew over the ground with a banner on display that read 'WENGER - OUT MEANS OUT!!'
Wenger's future remains uncertain, with the Frenchman still yet to announce whether he intends to stay on at the club beyond the current season.
But none of that seems to be distracting his team at the moment, with this representing a third win inside a week and fifth in six league games for a side who also have an FA Cup final against Chelsea to look forward to.
As they exerted early pressure at the Bet365 Stadium, Jack Butland did well to push Shkodran Mustafi's header over the bar in the ninth minute, and the visitors went even closer six minutes later when Nacho Monreal nodded against the post.
Stoke were struggling to get going but were presented with an opportunity just before the half-hour mark as Mustafi fouled Xherdan Shaqiri just outside the Arsenal box, picking up a booking in the process
The resulting free-kick was curled just over by Marko Arnautovic.
The Gunners were soon back on the front foot, with Sanchez firing narrowly wide after Glen Johnson lost the ball in a dangerous position.
And a fine Arsenal move then saw them grab the lead as Hector Bellerin ran on to Francis Coquelin's through-ball and delivered a cross that Giroud met with a tap-in.
It was another delightful Gunners move that made it 2-0 after the interval as Ozil gave the ball to Sanchez and then latched onto the superb pass the Chilean sent to him in return before beating Butland by lifting a shot over him.
That goal seemed to stir Stoke into life
Mame Diouf wasted a good chance by heading wide from Arnautovic's cross, the Senegalese forward's last contribution as he was then replaced by Crouch while Joe Allen was taken off for Saido Berahino.
Petr Cech then tipped a Bruno Martins Indi header over, before Crouch converted another Arnautovic cross in controversial circumstances, with it appearing the striker had used his arm to divert the ball in.
Wenger made his displeasure clear by the dugout, but the effort stood - and his men then responded admirably to the setback.
Sanchez put them 3-1 up as he received the ball from Bellerin and saw his strike deflect in off Ryan Shawcross.
Aaron Ramsey, booed by some Stoke fans having come on for Sanchez, then made a lively contribution at the ground where he broke his leg in 2010.
The Wales midfielder had one shot saved by Butland and subsequently provided the cut-back from which Giroud prodded in his second on the slide.
Source : PA