The CIS Insurance Cup winners might have been accused of overdoing the Hampden Park celebrations after a flat performance.
But Stephen McManus scored his eighth goal of the season, and Shaun Maloney curled home another trademark free-kick to move the Glasgow giants within touching distance of the title.
Celtic require eight points from as many games to guarantee the return of the trophy. But wins at Livingston on Sunday and against troubled second-placed side Hearts at Parkhead would wrap up the silverware early.
The home side should have been a few goals to the good before they eventually took the lead in the 36th minute from an unlikely source.
Nakamura swung over a corner and Stephen McManus sent a looping header over Brown and into the back of the net from nine yards.
Maloney curled another stunning free-kick past a rooted Brown from the edge of the area in the 79th minute to secure the points.
Substitute Dion Dublin saw his effort come back off the post late on but there was still time, however, for Boruc to suffer a major embarrassment when he let a long-range Hart effort slip under his body to gift Inverness a consolation.
: Boruc, Telfer, Balde, McManus, Wilson, Nakamura, Lennon, Petrov (Pearson 29), Maloney, Hartson (Dublin 68), Zurawski (Thompson 82).
: Brown, Duncan, Dods, McCaffrey, Hart, Munro, Hastings, Bayne (McAllister 79), Morgan, Black (Wilson 71), Wyness. Referee: C Richmond
: International Friendly: Germany 4 USA 1. Coca-Cola League Two: Boston United 0 Northampton 1. Spain La Liga: Alaves 1 Celta Vigo 0, Athletic Bilbao 1 Espanyol 1, Deportivo 2 Malaga 1, Osasuna 2 Real Sociedad 0, Real Betis 1 Racing Santander 0, Mallorca 1 Cadiz 0, Zaragoza 1 Real Madrid 1, Valencia 1 Villarreal 1. Italy Serie A: Cagliari 0 Fiorentina 1. Coppa Italia semi-final: Palermo 2 Roma 1, Inter 1 Udinese 0.