Bluebirds defender Gabor Gyepes fouled Boro striker Marvin Emnes and Arca sent David Marshall the wrong way with a left-footed penalty into the corner.
Craig Bellamy went close to a Cardiff equaliser but the Wales striker's shot shaved the crossbar and went over.
Michael Chopra then tested Boro keeper Jason Steele with a 20-yard effort.
Second-placed Cardiff had hoped to capitalise on Championship leaders Queen's Park Rangers' first defeat of the season at home to Watford on Friday night.
But, against a struggling Boro side who they had not lost to since 1984, Dave Jones' side continued their poor form.
They have now won just once in seven games, including last week's home draw to bottom club Preston
Meanwhile, boss Tony Mowbray enjoyed only his third win since replacing Gordon Strachan on 26 October as Boro claimed their first victory in five games.
Cardiff were without Championship top scorer Jay Bothroyd but the hosts were minus suspended centre-half David Wheater. And Boro's backline were put under early pressure when Chris Burke went close.
But former Swansea loan striker Emnes was quickly proving to be Cardiff's nemesis again as he forced Cardiff keeper Marshall into a smart save before strike-partner Scott McDonald poked harmlessly wide.
Emnes was the man the scored the south Wales derby winner for Swansea at the Cardiff City Stadium last month - and again the Dutchman had the beating of the Cardiff back-line.
Cardiff boss Jones bemoaned bad defending from Gyepes to allow Emnes space to score in the derby - and it was Gyepes again that was undone by the striker.
The Hungarian hauled down Emnes just inside the box and referee Anthony Bates pointed to the spot.
Gyepes protested his innocence but Argentine Arca worried little about that as he sent Marshall the wrong way to score his first league goal since September 2007.
Nicky Bailey's powerful drive was palmed away by Marshall as Boro looked for a second after the restart when Jones unleashed Chopra in a desperate bid to add much-needed impetus to Cardiff's attack.
And Bellamy fired a difficult first-time effort into the side-netting before clipping the top of the crossbar moments later with a fine drive.
Home keeper Steele then turned Chopra's chip to safety as Cardiff hoped to capitalise on the rest of the top five not winning either - but Boro held on for a morale-boosting victory , only their second home clean sheet this season.
Match Report courtesy of BBC Sport.
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