The Brazilian scored twice from a combined range of around three yards and added a sumptuous third following Lucas Moura's top-quality free-kick.
Alfie Devine was introduced at half-time to become Tottenham's youngest ever player and the teenager, at 16 years and 163 days, made a swift impact to drill home the fifth on the hour.
That completed the scoring at Rossett Park as eighth-tier Marine, who are 161 places below Spurs in the English league pyramid, inevitably succumbed against top-flight opponents.
Neil Kengni interrupted Spurs' early monopoly of the ball to crash a 35-yard dipping shot off the crossbar but, after surviving the heart-in-mouth moment, Dele Alli teed up Vinicius to jink past Bayleigh Passant and finish on the goalline.
Vinicius tapped in again on the rebound after Matt Doherty's side-foot volley from Alli's cross-field dink was parried into his path, and Lucas quickly made it three with a superbly executed free-kick.
More good work from Alli eventually led to Vinicius brilliantly looping home into the left side of the goal to complete his hat-trick.
3 - Carlos Vinicius is the first player to score an FA Cup hat-trick for a side managed by José Mourinho since Frank Lampard for Chelsea in January 2007 (v Macclesfield). Ruthless. pic.twitter.com/juYbD3PjdZ— OptaJoe (@OptaJoe) January 10, 2021
With the game won, Jose Mourinho introduced Devine for the second period and he flashed a fierce shot past Passant at the near post from Lucas' pass on the hour mark.
From there, Spurs were able to take their foot off the pedal a little and coast comfortably into the hat for round four.
What does it mean? Job done for Spurs
Save for Kengni catching Joe Hart unawares to rattle the woodwork, Spurs were never troubled and completed the job with routine professionalism. They comfortably came through their first tie against non-league opposition since 1995 when they defeated Altrincham.
Alli and Vinicius enjoy run-outs
Alli has fallen way down the pecking order under Mourinho and his long-term future at Spurs still remains in doubt. His class was on display here, though, with the midfielder finishing with an assist, five key passes and 85.7 per cent pass completion.
Vinicius too is having to show patience as back-up to Harry Kane, but for strikers scoring is always fun. He missed a late chance but scored with all of his shots on target in a clinical display.
No shame for Marine
The FA Cup always produces surprises but this was never likely to do so. Having said that, Marine – playing in the third round for the second time having gone out to Crewe Alexandra in the 1992-93 season – left it all out on the pitch against opponents of vastly superior quality and were a credit to the competition.
Tottenham have a decidedly tougher task at Aston Villa in the Premier League on Wednesday. Sadly for Marine, their domestic league remains suspended as a result of the coronavirus crisis.