An arrowed drive from Santi Cazorla after 31 minutes and a second by Tomas Rosicky after 62 minutes following a mistake from defender Danny Rose on the halfway line when he lost possession to Rosicky, secured Arsenal's place in the last 32.
Mainly restricting Spurs to half-chances and the occasional attacking break, Arsenal held out for a deserved win even though they played out the last few minutes with 10 men after Theo Walcott was carried off on a stretcher with a knee injury after Arsenal coach Arsene Wenger had used up his substitutions.
Walcott, who had a superb game at the centre of Arsenal's makeshift attack in the absence of current first choice strikers Olivier Giroud and Nicklas Bendtner, was involved in the game's most controversial incident when he was not even on the pitch.
As he was being stretchered away in front of the Spurs fans, he sat up on his stretcher and grinning broadly, gestured the 2-0 scoreline with his fingers, provoking an angry reaction from the visiting supporters who pelted him with coins.
Walcott's behaviour could lead to the FA investigating the matter although Wenger played it down afterwards.
"The doctor said to me that the coins were already raining over his head and they had to protect him before he made the gesture, but what he did was not offensive.
"I don't think the FA will take any action, I have seen the pictures and he is not being aggressive, he is smiling. I hope he is not banned, or does not miss any matches because of the injury," he told reporters afterwards.
Sherwood said he did not see the incident and preferred to concentrate on his own team's performance, which he admitted was "disappointing" after their recent good run following his permanent appointment two days before Christmas.
"But we've had six games in 17 days with probably the smallest squad in the league right now with all our injuries at the moment, and we showed great character, but we were fatigued and a little bit of sharpness was missing."
The Premier League leaders, who are seeking their first trophies in nine seasons since winning the FA Cup in 2005, were the better team in a typical blood-and-thunder north London derby despite the absence of strikers Giroud and Bendtner, midfielder Aaron Ramsey and defender Kieran Gibbs.
Walcott and teenaged German striker Serge Gnabry caused problems at the heart of the Spurs defence which looked vulnerable almost every time Arsenal poured forward.
Spurs, whose form has improved dramatically since Sherwood succeeded Andre Villas-Boas as coach, were always second best after a bright start, with Emmanuel Adebayor, who has played a key role in their recent good performances with four goals in five games, largely anonymous against his former club.
After losing 2-1 to West Ham in the Capital One (League Cup) while Sherwood was still caretaker, Spurs have taken 10 points out of 12 over the Christmas holiday period including a 2-1 win at Manchester United on New Years Day, but they never looked like getting much against Arsenal.
Their attention will now be focused on a top four finish in the Premier League, and progress in the Europa League, while Arsenal's quest for domestic or Champions League honours -- or both -- will be given a boost by this comfortable victory over their fiercest rivals.