The top-ranked British woman made a slow start and dropped the first set before suffering a further break of serve and deciding to take a medical time-out for cramp.
But the world No47 recovered from a difficult start to take the second set, before wrapping up the third in a blistering 34 minutes.
“I definitely expected myself to win today,” said Watson.
“I’ve had some tough draws in the past, and I knew it was a big opportunity today. I didn’t want to waste it.
“But at the same time I didn’t want to get frustrated with myself if some things weren’t going so well because that just wouldn’t help me out there on the court.”
Despite being confident of victory pre-match, Watson had lost to her Romanian opponent in their only other meeting, and the 21-year-old admitted she will need to improve in the next round against Kazakhstan’s Ksenia Pervak.
“Last time I played her [Pervak] was Australian Open juniors in the quarter-finals. I lost an she won that year,” added Watson.
“But, yeah, I’ve seen her around. A lefty and a tough player. She makes a lot of balls.”
Watson also played down concerns over her fitness after calling for the doctor and trainer in the second set, with the British No1 insisting that there were no issues surrounding her elbow, which had hampered her preparations for the tournament.
“No, my elbow was fine today,” she said. “It’s just because I haven’t been serving a lot. I didn’t really have my timing. I maybe felt it on one serve, but it was fine.
“I think I was struggling with the wind a bit at the beginning and my timing, but then I got into it.
“I just told them [the doctor and trainer] how I was feeling, if there was anything I could take, what I should be doing, because I was feeling quite faint. I didn’t have any energy today.”
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BIOGRAPHY: Unai Emery
BIOGRAPHY: Adam Lallana