Teenager Henri Lansbury - one of eight changes made by manager Arsene Wenger, who was serving a one-match touchline ban - swept the visitors ahead from close range after 15 minutes.
Robbie Keane, who came on at half-time, scored an equaliser three minutes after the restart to level for a much-changed Spurs side, which saw Brazilian midfielder Sandro make his debut.
However, neither team could find a winner in normal time, with Nasri's spot-kick double and Andrey Arshavin's goal sending the Gunners through.
There was a minute's applause before kick-off for Tottenham's double winning striker and England international Bobby Smith, who died on Saturday aged 77.
When the action started, there was the expected tense atmosphere of a derby, which Wenger watched from the stands, having accepted a misconduct charge and £8,000 fine by the Football Association following his actions at the end of the weekend's 1-1 draw at Sunderland.
Mexico forward Carlos Vela, who has scored three goals already this season, shot wide from 25 yards.
On nine minutes, Roman Pavlyuchenko was bundled over by Djourou out on the left.
The delivery from former Gunner David Bentley, though, was poor and flew over the crossbar.
Arsenal went ahead on 15 minutes. Tomas Rosicky - who had missed a crucial penalty up at Sunderland - worked the ball down the left to Jack Wilshere.
His ball back across the six-yard box caught the Spurs defence cold, and England Under-21 international Lansbury, who was on loan at Watford last season, stabbed in his first senior goal.
White Hart Lane was left in stunned silence, apart from the pocket of ecstatic travelling supporters behind the other goal.
The home faithful were on their feet, though, when Bentley dashed clear, but blasted his 20-yard effort over before Pavlyuchenko flashed an angled drive wide after cutting in from the left.
Tottenham made a double change at the start of the second half.
Republic of Ireland striker Keane came on for Giovani Dos Santos and youngster Jack Livermore was replaced by England winger Aaron Lennon.
There was some action on the Arsenal bench as assistant Pat Rice appeared to signal up to the stands before sending out Gael Clichy and Emmanuel-Thomas to warm up.
Moments later, though, and Redknapp's changes paid off as Spurs levelled after 49 minutes.
Kyle Naughton's pass caught the Arsenal defence square, as Keane raced into the centre of the area and his low shot squirmed under the arms of Lukasz Fabianski.
White Hart Lane was immediately transformed, as the home side went in search of a quick second.
However, Arsenal weathered the storm and had a decent spell, with Eboue and Vela both missing the target after good build-up.
Laurent Koscielny made a last-ditch block as Lennon cut into the right side of the Arsenal box.
With 18 minutes left, the Gunners sent on Marouane Chamakh, for Vela, and Arshavin replaced Rosicky.
Keane came close to snatching a winner when his header hit the base of the post, before Wilshere's stoppage-time free-kick caused panic in the Spurs six-yard box.
Arsenal snatched the lead again just 30 seconds after the restart when Sebastien Bassong tugged back Nasri in the box, and the Frenchman dispatched the resulting penalty.
The Gunners were awarded another spot-kick when Chamakh went down after cutting across debutant Steven Caulker, who tugged at his shirt, leaving referee Lee Probert little option and Nasri slotted his second penalty of the night into the bottom corner.
Tottenham replaced Sandro with Niko Kranjcar, before Bentley clipped the woodwork with a far-post header wide - much to the delight of the visiting fans.
Arshavin made it 4-1 when after being tripped by Kyle Naughton, he collected Wilshere's quickly-taken free-kick to dart into the area and slip the ball into the far corner.
Fabianski denied Keane with a brilliant reaction save from point-blank range after Nasri had cleared off the line.
The second period of extra-time was an instantly forgettable, but the jubilant Gunners fans revelled in every minute.