Unite said it will be the first time workers at the base on the Clyde in Scotland have taken strike action, warning that a walkout could have "significant implications" for the running of the site.
Over 95% of Unite members backed strikes on an 82% turnout in protest at a "derisory" 1% pay offer by employers Babcock Marine.
The union will now consider the ballot result and hold mass meetings before deciding whether to name strike dates.
Unite national officer Ian Waddell said: "Workers on the Clyde deserve better than the below inflation pay cut Babcock wants to impose on them and this overwhelming mandate for strike action is a reflection of their anger.
"This is a conscientious, highly skilled collective that understands the unique challenges of their working environment and have consistently worked with the employer down the years to help deliver sustainability for both Babcock and the Ministry of Defence on the Clyde.
"It is shameful that Babcock is trying to hide behind the MoD and say its hands are tied by the contractual arrangements, forcing a pay cut which is not only derisory but patently unfair in comparison with the consolidated above-inflation deals settled at Devonport and Rosyth.
"Put simply, we cannot accept regressive public sector pay policy being imposed on private sector workers and Babcock Marine should stand up for the loyal workforce."
The dispute involves workers ranging from cleaners to maintenance staff at Faslane and Coulport.