The 19-year-old stands 165 places behind the Romanian and, as such, was heavily tipped for an early exit.
But in front of a packed Court 14, set against the backdrop of a glorious opening day here at the All England Club, the young Briton put in an inspired performance to beat her opponent in straight sets, winning 6-2 6-2.
It was a calm and measured display from the rising star, who dominated with her serve and baseline strokes.
Contrary to appearance, Swan admitted that she had been ‘shaking’ with nerves throughout the match – but added that she was delighted to have come away with the win, which she secured in just 56 minutes.
“It was a good match,” she said. “I think I was pretty nervous at the start. Actually most of the way through I felt myself a bit shaky. I don’t think it affected my tennis too much. I’m so happy that I managed to come through.
“The last few weeks I’ve shown more myself than anyone I can compete with the top girls. Going into the match today with that mindset, I think that’s why I was able to win.”
The youngster added that she hopes to move up to a bigger court in the second round, where she’ll now face the winner of the first round clash between 29th seed Mihaela Buzarnescu and Aryna Sabalenka.
“I think the bigger crowd will help me actually. I played on Court 1 in Eastbourne last week, which is also a pretty big court. I like having the support behind me. I’m not going to be playing a British girl. It’s going to be fun, I think. I think it helps me.”
The 21-year-old Londoner shook off a nasty-looking leg injury, suffered during a slip in the second set, to push seventh seed Pliskova all the way but the Czech eventually prevailed 7-6 (7/2) 2-6 6-1.
Dart, who was handed wild-card entry at this year’s Championships, admitted afterwards she can take confidence from her performance and vowed to return next year by right.
Asked if she will set about securing a place at Wimbledon without needing a helping hand, 171st-ranked Dart said: “Yeah, definitely my short-term goal is to try to get my ranking up. That’s really what I try and focus on. Obviously I would love to be back here.
“This year I’ve been very fortunate that I had a wild card into this tournament. I was very excited. I thoroughly enjoyed my experience.”
Wildcard Broady, 24, held his own against the 13th seed in the first set before he was broken decisively in the 12th game. From there, the Canadian asserted himself to see out the rest of the match in style.