Ivanovic seemed headed for a straight set victory against Falconi, building a 7-5, 5-2 lead and, after letting that lead close to 7-5, 5-4, holding three match points on Falconi's serve. But the American fought them all off, won the set, held then brought up double break point for a 2-0 third set lead.
The No.5-seeded Ivanovic regrouped just in time, though, saving those break points and then reeling off six of the next eight games to close it out after two hours and 35 minutes, 7-5, 6-7(4), 6-3.
She may have had a close encounter in her opener, but Ivanovic absolutely loves the Miami Open.
"The tournament is great," she said. "Also the city is beautiful. We have so much sunshine and the beach, and this is the lifestyle I enjoy. I've been trying to use that for some relaxing down time."
How has the Premier Mandatory-level tournament changed over the years? "It's growing. There's so much more awareness about this event. And there are a lot of fans coming out - that's great to see."
Ivanovic's best performance at the Miami Open came in her first appearance in 2005, reaching the quarterfinals (falling to Amélie Mauresmo). She has reached the fourth round five times since then but hasn't gone further - can the former World No.1 get back into the final eight this fortnight?
Next up for Ivanovic is the No.27-seeded Lisicki, who was on court for even longer - two hours and 45 minutes - for a marathon 7-6(1), 6-7(5), 6-4 victory against fellow German Julia Goerges.
Ivanovic leads Lisicki in their head-to-head, 4-1. All five career meetings have come in the last 12 months, with Lisicki's lone win a hard-hitting three-setter in the third round of Wimbledon last year.
All four of Ivanovic's wins have come in straight sets, including a 6-0, 6-3 win at Dubai this year.
Ivanovic and Lisicki will square off at the Tennis Center at Crandon Park on Sunday.