Andre Claro put the visitors ahead in the 30th minute before Luis Martins received two bookings in quick succession shortly after the hour mark to earn himself a dismissal.
And Arouca took full advantage of their numeric superiority, as Roberto scored his side's second and another Claro effort made it three within eight minutes of the red card.
Gil Vicente have now gone five matches without a win - losing four - and sit eighth, while Arouca ended a nine-match winless run.
In the day's other fixture, struggling Belenenses held fourth-placed Estoril to a goalless draw, despite being reduced to 10 men late on.
Joao Meira went into the referee's book as early as the seventh minute and left his side clinging on for a point in the closing stages after he received a second yellow card in the 86th.
Despite picking up a point, Belenenses have now gone seven league matches without a victory and will want to turn their form around soon if they are to stave off the threat of relegation.
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