Premier League: Crystal Palace 1 Arsenal 2
The pair were on target before the interval in south London as Arsene Wenger's side moved third in the Premier League after Manchester United were beaten by Swansea City, with Southampton in action against Liverpool on Sunday.
A clumsy tackle from Pape Souare on Danny Welbeck allowed Cazorla to open the scoring from the spot after eight minutes, before Arsenal landed another blow on the stroke of half-time.
Quick thinking from Giroud inside the area allowed him to prod home the second and despite their resolute second-half display eventually being ended when substitute Glenn Murray scored late on for Palace and then struck the woodwork with another late chance, the visitors held on.
Arsenal - who were able to name Jack Wilshere in the squad for the first time since November after injury - were on the back foot early on as Palace threatened down the right through Wilfried Zaha.
They pressed Arsenal high up the pitch but sloppy play from Premier League debutant Souare gifted the visitors an opportunity to lead.
Referee Mark Clattenburg felt the foul on Welbeck took place inside the box and, after the official had pointed to the spot, Cazorla expertly slotted home.
It proved the only real chance of the opening 20 minutes, with Alexis Sanchez almost forcing an opportunity soon afterwards as he pressured Scott Dann into losing possession, only for Julian Speroni to come out and clear.
Palace, for all their promising spells of possession, struggled to test goalkeeper David Ospina and his defence before Arsenal made the most of their threat on the break when Giroud doubled their advantage.
A clever pass from Francis Coquelin found Welbeck and, when the England international's effort was saved, Giroud reacted quickest to register his 50th Arsenal goal in first-half stoppage time.
Mesut Ozil had Speroni worried with a looping header that flew over the crossbar early in the second half but, like in the first period, the game continued to see Palace have spells of possession against Arsenal's threat on the counter.
In a bid to get back in the game, home boss Alan Pardew introduced Yannick Bolasie from the bench for his 100th Palace appearance but, alongside the likes of Zaha, Jason Puncheon and Dwight Gayle, he failed to unlock the Arsenal defence.
Gayle could only angle Souare's header over the crossbar with 20 minutes left before Puncheon flashed a free-kick wide.
Substitute Murray - on as a 79th-minute replacement for Gayle - then almost single-handedly gained Palace a point, as he first poked home a shot inside the area before hitting the post with a diving header deep into stoppage time.
However, it proved a frustrating day for Pardew.