The claim not the first setback to hit the MoD scheme; it was revealed earlier this year that part of it was the subject of a fraud investigation.
The payment dispute arose over the non-payment of an interim invoice. Carillion referred this to adjudication, and in September adjudicator Gwyn Owen ordered Devonport to pay £7.45m, as well as fees of just under £14,000, by 12 October.
The writ says Devonport has refused to pay the money and that it does not intend to honour the adjudication. According to the writ, the dockyard is arguing that the adjudication is invalid and unenforceable.
Carillion has asked the High Court to order the payment of the sum, along with interest of £1636 a day.
Carillion's writ also alleges that Davenport has breached its obligations under construction contracts regulations, and has refused to pay the adjudicator's fees.
The writ says the two firms entered into an agreement and subcontract in March 1999 and Carillion carried out civil engineering, building and M&E work to develop a dry dock in Devonport for refitting nuclear submarines.
A statement from Devonport Royal Dockyard said: "The value of a contract associated with the recent nuclear facilities upgrade project at Devonport is the subject of ongoing discussion between Devonport, Carillion and the parties' advisers.
If a settlement is not reached then the matter will be settled through litigation. Devonport has no further comment on the matter."
In August it was revealed that contractors working on the project are at the centre of a multimillion-pound fraud investigation. MoD police are investigating allegations that labour was double-booked.
Building understands Carillion is not one of the firms facing investigation.
The eight-year project is the subject of an investigation by the National Audit Office after its costs increased from £417m to an estimated £679m.