The Gunners - who made sweeping changes from the midweek Carling Cup semi-final win over Ipswich - took the lead when Nicklas Bendtner's angled shot deflected off Terriers skipper Peter Clarke on 22 minutes.
However, influential midfielder Samir Nasri limped off with a hamstring problem, which could yet rule him out of the Champions League showdown with Barcelona, before Sebastien Squillaci was shown a red card by referee Mark Clattenburg for blocking Jack Hunt's surging run.
Alan Lee headed a deserved equaliser for the Terriers, but Fabregas won it from the spot after Jamie McCombe was adjudged to have pushed Bendtner over.
The Premier League title challengers were on top soon after kick-off.
Bendtner, who netted a superb goal to open the scoring against Ipswich in midweek, charged into the right side of the Huddersfield penalty area.
The ball eventually broke to Andrey Arshavin at the far post, but the out-of-sorts Russian blasted his shot high into the stands.
Arshavin continued to see plenty of possession, and cut in from the left before his low centre across the six-yard was turned just wide by Bendtner before his angled shot was pushed onto the outside of the post by veteran keeper Ian Bennett.
A stray pass from Squillaci suddenly set Kevin Kilbane clear, but as he played in Lee, Laurent Koscielny got across to cover.
Arsenal took the lead on 22 minutes via a deflection.
Bendtner collected the ball on his chest before charging toward the right side of the penalty area, where he fired in an angled drive which ricocheted off Clarke and flew into the bottom corner.
Huddersfield immediately went on the offensive and it needed a deft touch from Squillaci to poke the ball away from Lee as he looked set to net after a low ball into the Arsenal penalty area from the right by Anthony Pilkington.
There was concern for Arsenal when Nasri pulled up holding his hamstring after chasing a long drop-kick from Almunia.
Gunners boss Arsene Wenger was taking no chances, immediately replacing the French playmaker with Tomas Rosicky.
It should have been 1-1 on 38 minutes when former Aston Villa midfielder Joey Gudjonsson floated a high ball into the Arsenal penalty area, which eluded the centre-halves and dropped for Pilkington, but he headed wide.
There was then high drama when Squillaci was shown a red card, adjudged last man by Clattenburg, after he blocked Hunt following a surging run from the Terriers defender, which had started in his own half.
With no other defensive cover on the bench, striker Marouane Chamakh was sacrificed as Alex Song came in alongside Koscielny for the second half.
Arshavin was played clear by Rosicky down the left channel and took the ball into the area before stabbing it into the side-netting.
Arsenal were forced to be both patient and cautious as Huddersfield looked to get forwards themselves at every opportunity.
Arshavin made a brave sliding challenge at the near post to take the ball away from Gudjonsson as he shaped to shoot.
The Terriers refused to lay down as they were cheered forwards by their army of supporters packed behind Almunia's goal.
Jamie McCombe glanced a header just wide and then Almunia made a brilliant save at full stretch to push behind Lee's header, which look destined for the bottom-left corner.
The equaliser, though, eventually came on 66 minutes when Lee crashed in a bullet header from Pilkington's corner as the Clock End erupted in a sea of blue and white.
With 21 minutes left and Wenger desperate to avoid a replay, Abou Diaby was replaced by Fabregas, whose last-gasp spot-kick had kept his side in their third-round tie here against Leeds.
Bendtner sent a header just over as suddenly Arsenal found some fluency again.
The Gunners were handed a reprieve with five minutes left when Bendtner went down in the box under what looked minimal contact from McCombe, who received a yellow card.
Fabregas stepped up to send the keeper the wrong way and once again save his team's blushes.