The two sides had initially drawn 3-3 at Vicarage Road earlier this month and their replay was at first scheduled for 14 January, until heavy rain in the area meant it was rescheduled for this evening.
Playing on a weathered pitch and under heavy fog, Tranmere took the lead through Emmanuel Monthe before Watford equalised via Kaylen Hinds. A period of extra time was required and substitute Paul Mullin hit a winning goal for the hosts.
Tranmere's reward is a high-profile fourth-round tie against United at Prenton Park on Sunday, live on BT Sport in the UK.