The 21-year-old took his goal tally to four in three games as the Buddies booked their place in the last eight.
Goncalves headed home to break the deadlock six minutes after half-time and doubled the hosts' advantage eight minutes later.
Paul Heffernan's double sent Kilmarnock into the quarter-finals as Inverness suffered their second cup exit in eight days following this 2-0 result.
The substitute fired home in the 58th minute after Inverness were again exposed on the break, although they suffered an unfortunate run of the ball.
And the Irishman's clever back-heel with five minutes left ensured Caley Thistle have now lost as many games in those eight days as they had in the rest of the season, with their Scottish Communities League Cup semi-final defeat by Hearts followed by a league defeat at St Mirren.
- Sports & Recreation
- Paul Heffernan