The Togo striker, who apologised for his bust-up with team-mate Nicklas Bendtner during the 5-1 Carling Cup drubbing by Spurs in midweek, struck twice in the second half to end the brave resistance of Kevin Keegan's battling side - who are set to return here next week in the Premier League.
An own goal from Nicky Butt in the closing moments ended a frustrating afternoon for the visitors, who had enjoyed good spells, but just lacked a cutting edge in the final third.
Arsenal were eager to erase the memory of their drubbing at White Hart Lane in midweek when their arch rivals marched on to Wembley, and settled quickly.
However, Newcastle are a club rejuvenated under Keegan, and it showed as the visitors refused to give any ground during the opening exchanges.
The first opening came on five minutes, when Philippe Senderos' long ball into the penalty area almost picked out Fabregas, but was just too far ahead of the Spain midfielder who could not get any power on his shot at full stretch.
Arsenal were forced into an early change when winger Tomas Rosicky hobbled out and was replaced by Croatian hitman Eduardo.
Newcastle looked dangerous on the break - and they almost snatched the lead on 14 minutes when the ball dropped to Alan Smith on the penalty spot following a corner on the left, and his goalbound shot was headed off the line by Gael Clichy.
Arsenal, without key defender Kolo Toure because of the African Nations Cup, were anything but solid at the back as Newcastle captain Michael Owen dragged an effort across goal from the right.
The full-back - who has never scored for Arsenal - was in the thick of the action at the other end, when his curling shot from the left corner of the Newcastle penalty box was parried by Shay Given after 16 minutes.
The Newcastle keeper was alert again when full-back Stephen Taylor misjudged a high ball on the edge of the area, which let in Eduardo, whose angled drive was well saved at the near post.
Taylor almost made up for his error with a timely arrival at the other end, but his lopping header fell on top of the net.
The visitors were certainly in no mood to sit back - something of a trademark under Keegan. Damien Duff weaved his way into the Arsenal box, and had to be shut out at the near post.
Slowly Arsenal stepped up the pressure, with Adebayor flashing a header just wide under close attention from Newcastle centre-back Claudio Cacapa.
The hosts, though, were being denied space around the penalty area by a hard-working Newcastle side.
In first-half stoppage time, Abou Diaby cut inside from the left, and sent in a dipping effort from the edge of the box - which Given tipped over acrobatically.
Newcastle again looked determined to take the game to Arsenal following the restart but on 51 minutes, the Gunners were in front.
Eduardo made space for himself on the edge of the penalty area, and sent in a curling left-foot shot which had Given beaten, but struck the base of the keeper's right-hand post.
The defence failed to clear the loose ball, and in-form Adebayor showed good close control before lashing his effort into the left corner, to leave Arsenal celebrating their 100th goal at Emirates Stadium.
Newcastle could count themselves somewhat unfortunate to be trailing after their determined performance.
However, as the hour mark passed, the visitors did not let their heads drop and continued to press.
Owen felt he had a penalty shout when the ball was flicked up on to Senderos' arm, but referee Martin Atkinson thought otherwise.
Space was, though, now opening up in midfield for Arsenal, with Eduardo causing plenty of problems as he drifted out wide and then into the centre.
With 20 minutes left, it needed a fine saving block from Cacapa to deny the Croatian, after he collected a flick into the six-yard box and rifled in his first-time effort on the turn.
Arsenal made sure of their place in the last 16 when Adebayor darted into the right side of the area, held off two defenders and drew the keeper, before firing in a low shot which last man Taylor could not keep out as the Togo hitman closed in on 20 goals of an impressive campaign.
An own goal from Butt made it 3-0 on 82 minutes, as Eduardo looked to nod in Fabregas' free-kick from six yards.
Any reproduction, publication or redistribution of this material without the written agreement of 365 Media Group is strictly forbidden.