The Englishman, 45, reeled off six straight legs to defeat De Zwaan 6-5, surviving three match darts along the way.
“It was a weird game. I’d rather start the match in pieces than end it,” joked Chisnall, who faces former European Champion Ross Smith on Saturday.
“That last leg was a bit weird; we both didn’t score well. But I took the finish, and I’m happy."
“I’ve got to play a lot better tomorrow than I did tonight. Ross Smith is a great darts player," Chisnall added.
“He’s beaten me in seven of the last seven games, but Michael van Gerwen had beaten me 26 out of 26 games, and I did get one win eventually.”
Menzies was beaten 6-4 by Paul Krohne in their first-round meeting, with the German holding firm despite a late rally by the Scot to claim his first career European Tour win and set up a tie with Josh Rock.
Elsewhere, Ireland’s Keane Barry suffered a 6-1 defeat to Niels Zonneveld, who will face three-time champion Van Gerwen in round two, as English pair Andrew Gilding and Ricky Evans endured contrasting fortunes.
Gilding lost 6-3 to Lithuania’s Darius Labanauskas, but Evans progressed with a comfortable 6-0 win over German Florian Preis.
England’s Daryl Gurney and Joe Cullen later came through tests against Polish duo Sebastian Bialecki 6-4 and Wojciech Brulinski 6-3 respectively, with Cullen now set to face Gerwyn Price. Their compatriot Luke Woodhouse beat Germany’s Dragutin Horvat 6-0.
Ireland’s William O’Connor lost 6-2 to Austria’s Mensur Suljovic, while Dutchman Wessel Nijman fought back from 3-0 down to beat Spain’s Cristo Reyes in a last-leg decider to round off the day.
Earlier, Dutchman Kevin Doets produced the standout performance of the opening session, averaging 117.12 -- the highest ever recorded at the European Darts Trophy -- in a 6-0 whitewash of Finland’s Jonas Masalin.
“That was good, right?” quipped Doets, who will face Cam Crabtree in round two on Saturday evening.
“I don’t know what happened, I just felt very comfortable."
“I started to feel comfortable when Jonas missed those darts for 1-1, but I don’t expect Cam to miss those chances tomorrow," Doets added.
“I’m in a really good place right now, I’m very confident, and I think I’m going to win tomorrow as well.”
Results
Afternoon session
Darius Labanauskas 6-3 Andrew Gilding
Kevin Doets 6-0 Jonas Masalin
Ricky Evans 6-0 Florian Preis
Niels Zonneveld 6-1 Keane Barry
Paul Krohne 6-4 Cameron Menzies
Richard Veenstra 6-3 Florian Hempel
Krzysztof Ratajski 6-4 Shane McGuirk
Lukas Wenig 6-3 Ryan Joyce
Evening session
Dave Chisnall 6-5 Jeffrey de Zwaan
Dirk van Duijvenbode 6-2 Mario Vandenbogaerde
Luke Woodhouse 6-0 Dragutin Horvat
Daryl Gurney 6-4 Sebastian Bialecki
Joe Cullen 6-3 Wojciech Brulinski
Niko Springer 6-5 Kai Gotthardt
Mensur Suljovic 6-2 William O'Connor
Wessel Nijman 6-5 Cristo Reyes