Ferrer had a chance to break serve at 0-0 in the fourth and four more break points at 3-3, but Ancic averted the mini crisis and overpowered Ferrer in the tie-break.
Ancic, who has now reached the fourth round on his last four appearances at Wimbledon, would play either defending champion Roger Federer or Lleyton Hewitt in the quarter-finals were he to beat Verdasco.
Ancic missed last year's event because of glandular fever, which forced him to miss six months of 2007 and was the last man to defeat Federer on grass, sending the Swiss crashing out in the first round on his debut in 2002.
"I've been spending hours in the gym dreaming of playing on Centre Court again so this is a big moment," Ancic told BBC Sport.
"I have great memories of Wimbledon and this was an incredible match, we were both fighting until the end. Ferrer showed what a great player he is."
Safin came out on top in a war of attrition with Italian Seppi.
Safin's best Wimbledon was a quarter-final spot in 2001
The Russian took the first and third sets, in which neither player dropped their serve, on tie-breaks but the second was a real contrast with five breaks between the pair.
Seppi broke first to lead 2-0 and although Safin replied immediately at 2-1, the Italian reeled off three straight games as Safin momentarily lost his way.
Safin's response was to go for broke and it brought him back from 5-1 to 5-3 before Seppi regained his composure and served out.
The fourth set was another tight affair with Safin breaking to lead 3-1 and he went on to wrap up victory.
His next opponent in the fourth round on Monday is Swiss 13th seed Stanislas Wawrinka, who had the easiest passage into the last 16 when Germany's Mischa Zverev retired while 7-5 6-1 down.
Safin told BBC Sport: "We couldn't see ourselves on court and it was freezing cold. But the crowd stayed until the end which was great.
"It was a tough match, especially at the end as I was getting nervous but it was also a great match, I think. There's no pressure on me and it's just match by match from now on."
Verdasco, Ancic's next opponent upset the Wimbledon seedings by beating Tomas Berdych 6-4 6-4 6-0 in the third round.
The 22nd-ranked Spaniard prevailed over the 11th seed from the Czech Republic.
Joining him is Cypriot 10th seed and former semi-finalist Marcos Baghdatis, a 7-6 (7-2) 6-4 6-2 winner over German qualifier Simon Stadler.
He now faces Feliciano Lopez after the Spanish 31st seed beat American Bobby Reynolds 6-4 7-5 4-6 6-4.
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