Mooney admitted he faked Ms Inah's documents so she could work at the couple's home in Coulsdon, Surrey, for 11 hours a day, five days a week.
She also cared for their three year-old son and was paid just £300 a month.
Both Mooney and Vukic had worked as immigration officers in Lagos, before Mooney took up his management position with the Foreign Office in the Nigerian capital.
Prosecutor Riel Karmy-Jones had told the jury: "The motive is one of greed. Had they succeeded, had Rose come to the UK for a period of a year, she would have worked for what can only be described as a pittance."
She said the plan was hatched in May 2011 at a secure compound where government staff were posted.
"They hit upon a plan that their Nigerian nanny and maid Rose would be the ideal candidate to continue to look after their son when Natasha worked in London."
Mooney submitted the form on May 25 last year, together with a copy of the maid's contract claiming she had been working for the couple since May 10, 2010.
Miss Karmy-Jones told jurors that Ms Inah had only been working for the couple since November 2010 - six months less than they had written on her visa application.
Their application attracted the attention of visa issuing officer Lindsey Moore, who knew the couple and recalled a visiting their flat in the autumn of 2010 - and at the time the couple were annoyed at another unreliable maid they had employed.
Mrs Moore is then said to have heard that the couple had sacked the unreliable maid shortly afterwards and found a replacement called Rose - months later than claimed on the application form.
An interview with the former maid, and Mooney's staff emails confirmed the date of the Miss Inah's appointment.
The prosecutor claimed that the cost of child care in London had been an issue when the couple were transferred back to the UK.
Mooney, of St Andrews Road, Coulsdon, Surrey, admitted assisting unlawful immigration, while Vukic, of the same address, was cleared of the charge.