The first episode of the 13-part series was watched by an average audience of 615,000 viewers and a 3.4% share on multichannel station Five US, according to unofficial overnight figures.
The series stars 'Friends' actress Cox as a manipulative, but troubled, tabloid magazine editor. It peaked at 653,000 viewers and a 3.6% share during the second half of the one-hour show in a 9pm slot.
'Dirt' won its multichannel slot and also helped Five US to the largest daily share of the multichannel stations, with a 1.3% share.
The series, which is sponsored by cosmetics brand Simple, is repeated at 11pm on Tuesday nights on Five US.
Fox International's FX channel also experienced a boost, scoring its biggest ever audience in the UK with the opening episode of US forensics drama 'Dexter' on Sunday.'Dexter' captured 300,000 viewers, giving it a 3.6% share of the pay-TV audience, for its debut outing at 10pm.
This gave FX the lead for its timeslot in cable and satellite homes. FX also came top among men in multichannel homes and the show gave the channel its biggest audience to date.
'Dexter' stars 'Six Feet Under' actor Michael C Hall, as a Miami forensics specialist who leads a double-life as a serial killer under the alias Patrick Bateman -- the name of the Huey Lewis and News-admiring psychopath in Brett Easton Ellis' 'American Psycho'.
FX launched its largest cross-media campaign to date for 'Dexter', spending a seven-figure sum on TV, press, cinema, internet and mobile ads.
Channel 4's 10pm instalment of 'Big Brother' attracted 3.6m viewers and a 21% share, giving the reality TV series the largest share of audience during the second half of the show.
ITV1's ratings flop, 'Tycoon', has shed more viewers since its debut last month. The show was hovering just below the 2m viewers mark during the first two episodes but appears to have settled around the 1.5m mark. Last night's instalment in a 10pm slot drew 1.5m viewers and a 6.9% share.