Christian Eriksen was played in one-on-one with Lukasz Fabianski early on but saw his shot saved.
Theo Walcott also had chances before Cazorla put the Gunners ahead on 31 minutes after being teed up by Serge Gnabry.
Rosicky then doubled the advantage from a Spurs corner after dispossessing Danny Rose on the halfway line and running into the box before chipping the onrushing Hugo Lloris.
Tottenham, who had shown signs of improvement under new boss Tim Sherwood, were overrun in midfield as their strike pair Emmanuel Adebayor and Roberto Soldado failed to cause many problems.
The victory came at a cost, however, with England forward Walcott carried off on a stretcher in the closing stages after appearing to suffer a knee injury.
It was a bright start by both sides.
Spurs created the first opening on nine minutes as a deflected clearance sent Eriksen in on goal.
However, Fabianski spread himself well at the near post and put the ball behind for a corner.
Arsenal then broke quickly as Gnabry fed Walcott, and he took his shot early from 20 yards, which was bouncing into the bottom left corner before Lloris touched it wide.
Walcott continued to stretch the Spurs defence, with a curling effort deflecting off Vlad Chiriches and just wide before, from the resulting corner, Gnabry fired a shot inches over.
On 18 minutes, Arsenal captain Thomas Vermaelen, recalled to the heart of the defence, upended Mousa Dembele and was booked by referee Mark Clattenburg.
Cazorla then whipped a 20-yard effort only a foot off target before Walcott beat the offside trap, but his low shot was kicked away by Lloris.
Arsenal took the lead on 31 minutes after Gnabry collected Bacary Sagna's pass and drove at the Spurs box. As Kyle Walker came across to cover Michael Dawson, the German stabbed the ball out to Cazorla on the overlap down the left and the Spaniard unleashed a fierce drive into the net.
Spurs looked to rally with Aaron Lennon again the main outlet.
However, they could well have been down to 10 men when Dembele pushed the back of his head into Jack Wilshere's face as the pair tussled following a coming-together in the centre-circle. Despite appearing to be close to the incident, referee Clattenburg took no action.
After some positive Spurs possession at the start of the second half, Adebayor failed to connect properly with Lennon's cross into the Arsenal penalty area - much to the amusement of the home fans in the Clock End.
On the hour, Walcott cut into the right side of the Spurs box and rounded Lloris, but the angle was too tight and his cut-back hit the side-netting.
It was 2-0 soon afterwards, though. After Arsenal had cleared a Spurs corner, Rose dawdled on the halfway line, which allowed Rosicky to nip in and take the ball from the Spurs defender.
The Czech playmaker sprinted clear and on into the penalty area, where he held off Walker to dispatch a lovely chip over the keeper and send Arsenal into round four.
Fabianski held onto a low drive from Dembele, before Arsenal brought on Mathieu Flamini for Wilshere and Mesut Ozil for Mikel Arteta.
Walcott was unfortunate again to see a low effort curl just wide from 15 yards after more defensive lapses.
Spurs felt they should have been given a penalty when Walker flicked the ball up onto Cazorla's arm, but neither the referee or his near-side assistant Peter Kirkup were interested.
Arsenal had to play the closing stages with 10 men as Walcott was carried off on a stretcher after appearing to jar his knee.
The forward was taken around behind the Arsenal goal, in front of the Spurs fans - some of whom threw objects.
Walcott signalled a "2-0" gesture with his hands, before donning an Arsenal scarf as he was carried on around the stadium.