French heavyweights Stade Francais had already wrapped up qualification from Pool 4 into the knock-out stages but Leeds finished as runners-up with four victories to their name.
Two Florin Vla penalties gave the Romanians a surprise lead at the break before Leeds rallied in the second period, running in four tries.
Vla kicked his side ahead with two first-half penalties before Semi Tadulala crossed for the opening try of the afternoon. Leigh Hinton could not add the extras and Bucharest went into the break 6-5 ahead.
Leeds captain Marco Wentzel led the fightback, touching down before Miguel Alonso scored his first try for the club. Chris Lewis-Pratt converted both scores to extend Leeds' advantage.
Luther Burrell grabbed the bonus-point score for Leeds.
In the other game in Pool 4, Stade Francais beat Crociati Rugby 34-17 in Italy to become the highest-ranked team in the quarter-final draw.