The Falcons ran in six tries before half-time, with Will Witty opening the floodgates in the very first minute.
Full-back Lee Smith added a second soon after, centre Chris Harris crossed the whitewash next and, after Florin Ionita gave Bucuresti some brief hope, Mark Wilson joined in the fun before Tom Penny went over twice in 11 minutes.
The second period failed to live up to the same high standards, with Ionut Botezatu salvaging some pride for the visitors with a brace of tries.
However, David Nelson had the last word when touching down for Newcastle in a convincing victory, with Rory Clegg kicking three conversions and Smith another.
There was late drama in a tight encounter in Bordeaux-Begles as Edinburgh came from 13-5 down with 10 minutes to play to prevail 15-13.
The only points in the first half came from the boot of Lionel Beauxis as the home team led 3-0 at the break.
The visitors started the second period brightly when winger Tim Visser went over but scrum-half Baptiste Serin replied for Bordeaux and Beauxis successfully converted for a 10-5 advantage on 50 minutes.
Beauxis added another penalty as the French side threatened to take control as outside half Phil Burleigh was sent to the sin bin, but substitute Tom Heathcote kept Edinburgh in it with a 70th-minute shot at goal.
Spurred on by that score, the visitors pushed forward and were dramatically awarded a penalty try shortly after, which Heathcote converted for a gutsy two-point win.