On the eve of the contest, Bellamy referenced his side's woeful away record in friendly fixtures.
Wales are now without a victory in 17 away friendly matches - a run that stretches back to November 2008, when Bellamy himself scored the only goal of the contest to earn his side a 1-0 triumph over Denmark.
He called for dominance in Bucharest, and in terms of controlling possession, the head coach got his wish.
Wales fully dictated the opening stages, with captain Ethan Ampadu, Brennan Johnson and Kieffer Moore all having early chances.
But they were fortunate not to be punished for failing to make the most of those opportunities when - shortly after Tudor Baluta drilled at Danny Ward - Munteanu unleashed a piledriver into the net, only to be wrongly denied by the offside flag.
Ward then recovered from his own error to tip over another Munteanu effort before Nicolae Stanciu curled a free-kick agonisingly wide to bring an end to an entertaining opening half.
Daniel James and Coman both had efforts early on in the second half, although the latter made his mark just seven minutes after coming off the bench.
The hosts attacked Wales' right flank, and Darius Olaru picked out Coman who had space around 10 yards out to coolly slot beyond Ward.
Wales needed a spark, and Bellamy sent on Connor Roberts, Lewis Koumas and Brooks with an hour gone.
The changes reaped instant rewards as James charged up the left flank, and his floated cross found Brooks unmarked at the back post as the Bournemouth ace levelled proceedings with his seventh international goal.
After pegging back Hagi's men, Wales pushed for a winner, with Tarnovanu denying Koumas and Cameron Congreve in quick succession.
But the hosts found the net for a second time as Rus nodded Matei's delivery into the net.
After Matei then struck the post, Koumas was superbly denied a second last-gasp equaliser in the June international camp as Rus blocked his goalbound shot.
That ensured the men in yellow were able to celebrate after earning Hagi his first victory since being appointed as Romania boss in April - leaving Wales to rue their missed opportunities.
