Tottenham had won at Manchester United in their previous outing and new manager Tim Sherwood made only one change to his starting XI, but his side fell behind in the 31st minute at the Emirates Stadium.
German teenager Serge Gnabry was the architect, picking the ball up wide on the Arsenal right and driving infield before finding Santi Cazorla, who speared a left-foot shot past Hugo Lloris.
Spurs were unable to react and Tomas Rosicky sealed victory in the 62nd minute, robbing Danny Rose on halfway and bearing down on goal before adroitly lifting the ball over the advancing Lloris.
The game ended on a sour note for Arsenal when forward Theo Walcott had to be stretchered off after falling awkwardly, but he showed that he remained in high spirits by goading the visiting fans as he was carried away.
"The game was played at a good pace and we controlled both sides of it quite well, the defensive and offensive," said Arsenal manager Arsene Wenger.
On Walcott, whose '2-0' gesture as he was stretchered off prompted a hail of missiles from visiting fans, Wenger added: "It does not look aggressive. He was smiling."
Last season's beaten finalists Manchester City tasted fresh disappointment in the competition when they drew 1-1 at second-tier Blackburn Rovers.
Stunned by Wigan Athletic in last year's final, City took the lead at Ewood Park through an Alvaro Negredo goal on the stroke of half-time, only for Scott Dann to equalise early in the second period.
City, currently second in the Premier League, also had defender Dedryck Boyata sent off late on for two bookable offences and now face an unwelcome replay against Gary Bowyer's side.
"I think it was a very close game," City manager Manuel Pellegrini told BT Sport.
"They're a difficult team and they play here with a lot of intensity, so we must decide who continues in the FA Cup at home."
Villa became the first major casualty of the third round after losing 2-1 at home to League One representatives Sheffield United.
Nicklas Helenius appeared to have kept Villa in the competition when he cancelled out Jamie Murphy's deflected opener in the 75th minute, only for Ryan Flynn to give the Blades victory six minutes later.
"For us and our 6,000 fans who came today, I think you saw how much it meant," said winning manager Nigel Clough.
"It's still a special, special competition and we'd just like a home draw next because we've had three away games on the spin, so it would be nice to get one at Bramall Lane."
Former Manchester United striker Ole Gunnar Solskjaer tasted success in his first game as manager of Cardiff City after his new side came from behind to win 2-1 at Newcastle United.
Solskjaer, who succeeded the sacked Malky Mackay on Thursday, saw his new charges fall behind to a 66th-minute Papiss Cisse strike, but two goals in seven minutes from substitutes Craig Noone and Fraizer Campbell gave Cardiff victory.
"It's not about me, but it is a perfect start to these new times," Solskjaer said.
"I think the lads have had a lot to deal with lately and it's nice for them to start with a win in the cup."
In the day's two other all-Premier League ties, goals from Dwight Gayle and Marouane Chamakh gave Crystal Palace a 2-0 win at West Bromwich Albion and Fulham drew 1-1 at Norwich City.
Everton crushed Championship high-fliers Queens Park Rangers 4-0, with Nikica Jelavic claiming a brace, while Southampton edged Burnley 4-3 in an entertaining game at St Mary's.
Hull City won 2-0 at Middlesbrough and Stoke City defeated Championship leaders Leicester City 2-1.
Wigan's trophy defence began in uncertain fashion as they were held to a 3-3 draw by third-tier Milton Keynes Dons, while Millwall, semi-finalists last year, crashed out after an embarrassing 4-1 defeat at fourth-division Southend United.
Bolton Wanderers belatedly exacted revenge for their 4-3 loss to Blackpool in the 1953 final by beating their Championship rivals 2-1, while Leeds United fell to a shock 2-0 defeat at Rochdale.
Meanwhile, the lowest-ranked team in the tournament, non-league Macclesfield Town, claimed a 1-1 draw at home to second-tier Sheffield Wednesday thanks to a 72nd-minute equaliser from Steve Williams.