Hooker Vano Karkadze and halfback Mikheil Alania also crossed for Georgia, who have provided some immediate success for new coach Richard Cockerill following his appointment in January.
Portugal scored a consolation try late on through Rodrigo Marta but struggled with Georgia's physicality as their scrum wilted and they lost numerous contests at the breakdown.
They were unable to match their exciting performance at the Rugby World Cup in France last year, when the teams played to a hugely entertaining 18-18 draw in Australia's Pool C.
Sunday's final at Stade Jean Bouin in Paris was played at a frenetic pace with little structure from either side as Georgia led 12-3 at halftime.
They scored the opening try after 43 minutes when five-eighth Luka Matkava's cross-field kick was perfect for Tabutsadze to dot down in the corner, while their second was more brute force as the forwards rumbled the ball over the line via Karkadze.
By then the game was up for Portugal, and Tabutsadze showed a clean pair of heels to race in for the third try before Alania finished off a flowing move as Georgia picked holes in their opponents' defence.
Portugal, who were stunned 10-6 by Belgium in the pool stages, now look forward to a fixture against world champions South Africa in Bloemfontein on July 20.
Spain took third place, holding off a late surge from Romania to win 40-33.
