Spain midfielder Cazorla opened the scoring with a fine left-footed strike in the first half to double his goal tally for the season.
Tottenham has been resurgent under new boss Tim Sherwood, but was second best on Saturday and Rosicky took advantage of an error from Danny Rose to give Arsenal breathing space in the second half.
The visiting side was unable to recreate the performance that saw it beat Manchester United 2-1 at Old Trafford on New Year's Day as the club's four-game unbeaten run came to an end.
The Premier League leader has not won a major trophy since 2005, but victory over its neighbor ensures that the club is still in contention for honors on three fronts.
Arsenal boss Arsene Wenger made four changes to the side that beat Cardiff City, with Lukasz Fabianski, Thomas Vermaelen, Rosicky and Serge Gnabry coming in.
Sherwood, who played in the last FA Cup tie between the two sides at the semifinal stage in 2001, gave a first start to teenage midfielder Nabil Bentaleb as fellow Frenchman Etienne Capoue dropped to the bench.
Denmark international Christian Eriksen had a great chance to open the scoring after only eight minutes, but Fabianski denied him at the near post after the midfielder had raced into the penalty area.
A sell-out crowd generated a lively atmosphere and Theo Walcott, playing as a striker, almost raised the decibels further when his long-range strike was tipped wide by Hugo Lloris 11 minutes in.
Walcott was a constant threat and two minutes later his measured left-footed effort deflected just wide, then Gnabry shot just over the crossbar a few moments later as the game burst into life.
England international Walcott had his first clear-cut chance when he went one-on-one with Lloris after 24 minutes, but the France goalkeeper saved with his feet.
Arsenal looked the more likely to take the lead and was in front 31 minutes in, when Gnabry picked out Cazorla with an incisive pass and he fired home first time with his left foot from inside the penalty area.
Former Arsenal striker Emmanuel Adebayor turned sharply in the penalty area early in the second half but lost his footing as he was about to shoot, much to the delight of the home supporters.
Gnabry was a thorn in the side of the Tottenham defenders and the tricky winger picked out Walcott with a clever pass, but the former Southampton man found the side-netting from a tight angle after going around Lloris.
Arsenal was gifted a second goal after 61 minutes though, when Rose was robbed by Rosicky on the halfway line and the Czech Republic midfielder raced clear and beat the advancing Lloris with a composed finish.
Tottenham looked disjointed at the back and Walcott came close to adding a third goal, but he curled a right-footed effort just wide and was then carried off on a stretcher - not looking in too much discomfort after suffering an injury eight minutes from time.
Lloris produced a fine save to deny substitute Mesut Ozil a third goal after 89 minutes, but Arsenal comfortably progressed to the fourth round.