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The world No. 75 unsettled the hard-hitting Puchkova, who defeated Venus Williams in the semi-finals, with a range of slices, angles and drop shots.
Niculescu, who saved three match points against Spaniard Anibel Medina Garrigues in her first round match when she was 6-3 down in a second set tie-break, said she was hugely relieved to have finally won a WTA Tour title after two previous losses in finals.
"I competed in two finals before and lost to (Victoria) Azarenka and Venus, and today I finally managed to win one - I'm so happy," she said.
"I felt so tired on the court but really, really wanted to win, and right now I can't really explain how happy I feel. I'll just relax tonight and go to bed, and tomorrow I'll think about what I've done here."