The day didn't start well for Dellacqua, who was late to the court, with this minor hiccup seeming to frazzle the 30-year-old.
She went down three games in seven minutes, making 14 unforced errors in the opening phase of the match.
However, Dellacqua steadied and came back to claim the first set 7-6 (3).
It was all smooth sailing in the second however, as the Australian stuck to her gameplan, and hit the ball hard when balanced, closing out the set 6-3 with some impressive power hitting.
@caseydellacqua through to the 3rd round with a win over Svitolina @Wimbledon #youbeauty #gothegirls @TennisAustralia pic.twitter.com/ASBYfB1AyB
— rennae stubbs (@rennaestubbs) July 2, 2015
Elsewhere, Petra Kvitova continued her march toward a repeat championship at Wimbledon on Thursday. Madison Keys of the United States also was among the winners, but Christina McHale could not hold on for an upset victory.
Carolina Wozniacki also advanced to the third round, which starts Friday. Day 4 stars in action include Serena Williams, Maria Sharapova and Americans Venus Williams, Bethanie Mattek-Sands, Sloane Stephens, Coco Vandeweghe and Madison Keys.
Observers didn't see much of a challenge to Kvitova, as the two-time champion dismissed diminutive Kurumi Nara 6-2, 6-0.
Next for Kvitova: former world No. 1 Jelena Jankovic.
Thursday was similar to Kvitova's first match, in which her opponent won one game.
Other Thursday results
— No. 21 Keys continued to her week with a 6-4, 7-6 (3) victory over Elizaveta Kulichkova. Coached by Lindsey Davenport, Keys has yet to perform to full expectation but kept American women among the winners.
— No. 28 Jankovic needed three sets to beat Evgeniya Rodina 6-7 (4), 6-1, 6-3.
— Kristyna Pliskova knocked out No. 26 Svetlana Kuznetsova 3-6, 6-3, 6-4.
— No. 18 Sabine Lisicki beat McHale 2-6, 7-5, 6-1.
— No. 5 Caroline Wozniacki advanced with a 6-1, 7-6 (8) victory over Denisa Allertova.
Other seeded winners included Angelique Kerber (10), Agnieszka Radwanska (13), Timea Bacsinszky (15), Garbine Muguruza (20) and Camila Giorgi (31).
Two other seeds were eliminated: Ekaterina Makarova (8) and Alize Cornet (25).
Friday's matches
The women's headliner comes on Centre Court, where Serena Williams faces Britain's Heather Watson in the day's third match.
After Stephens faces No. 6 Lucie Safarova in the day's second match, No. 16 Venus Williams meets Aleksandra Krunic on Court 2.
Court 3: Vandeweghe faces No. 22 Samantha Stosur of Australia in the second match.
Court 18: Mattek-Sands faces No. 30 Belinda Bencic in the day's second match.