The Lombard Street venue is one of 22 Caffè Nero branches in Belfast.
The firm has a total of 40 outlets across Northern Ireland and nearly 700 shops throughout the UK and Ireland.
Barrister for defendant company Nero Holding Ltd, Chris Sherrard, detailed a series of steps taken since the inspection.
“Mesh has been inserted into the drain that’s thought to be the cause for concern,” he said.
Mr Sherrard argued that the branch is located in a congested and problematic part of the city for pest control.
Passing sentence, Judge Conner said: “Whilst I want to give credit for the plea, these are very serious matters.
“I will impose a fine of £2,000 in respect of each of the two complaints.”
Caffè Nero was also ordered to pay the Council’s £359 legal costs in the case.