Mata has struggled to convince Mourinho he is worthy of a regular place in his team and the Spanish midfielder has started just six games this season.
But he was back in the starting line-up at the Emirates Stadium and made the most of his opportunity with an influential display capped by a fine second half strike which booked Chelsea's place in the quarter-finals.
Mata's first club goal of the season came after Blues defender Cesar Azpilicueta had opened the scoring before the break.
Chelsea, twice League Cup winners under Mourinho, were always in control despite 10 changes to the team that beat Manchester City 2-1 on Sunday.
Mourinho had been incensed that Chelsea were forced to play twice in just three days, claiming that Arsenal had been given an unfair advantage because their weekend fixture against Crystal Palace took place at Saturday lunchtime.
He insisted he would send out a weakened team in protest and was true to his word as only Gary Cahill retained his place, although the strength of Chelsea's squad was clear to see as the likes of Mata, Willian and Samuel Eto'o hardly constituted a significant drop in class.
Arsene Wenger had hoped Arsenal would be able to inflict a psychological blow on one of their main challengers for their current position on top of the Premier League.
But Wenger made eight changes from the win at Palace and his stand-ins can have no complaints if they go back to the bench after a disjointed performance.
Mourinho and Wenger had a spikey relationship at the best of times during the Chelsea manager's first stint in England, but their relationship seems to have thawed a little judging by the warm smile and handshake the old rivals exchanged before kick-off.
That good-natured exchange set the tone for a surprisingly tame opening to a usually frantic London derby.
Yet with Arsenal struggling to find any rhythm, it wasn't a surprise that the opening goal in the 25th minute came from a miscue in the Gunners rearguard.
After clearing a corner, Chelsea broke at speed with Eto'o leading the charge before Aaron Ramsey's challenge on Michael Essien sent the ball looping towards the edge of the Arsenal penalty area.
Carl Jenkinson and Jack Wilshere hesitated over who should clear and when Jenkinson finally took action his attempt to head back to Lukasz Fabianski lacked both the power and direction to reach the Gunners goalkeeper.
Azpilicueta took full advantage of the gift, lunging in bravely to poke the ball past Fabianski into the far corner.
The Chelsea right-back nearly returned the favour when his careless play moments later let Nacho Monreal in for a low strike that flashed just wide.
That was Arsenal's only serious chance of the half, which spoke volumes for the way the Gunners were hamstrung by Nicklas Bendtner's lack of sharpness in a rare start for the Danish striker.
Eto'o wasn't far away from increasing Chelsea's lead with an opportunistic long-range strike early in the second half, prompting Wenger to send for the cavalry in the shape of German playmaker Mesut Ozil, who replaced ineffective Japan winger Ryo Miyaichi.
But before Ozil had a chance to make an impact, Mata had put the game beyond Arsenal with a brilliant strike in the 66th minute.
A loose ball fell to Mata on the edge of the Arsenal area and the Spaniard took one touch to control with his left foot before unleashing a ferocious right-foot strike that bent away from the despairing dive of Fabianski.
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