In an all-Italian affair at the WTA International event, second seed Vinci was eventually beaten by Camila Giorgi 6-3 6-7 (7-9) 6-0 in two hours and four minutes.
Vinci, the 2013 champion, fought hard to stay in the clash in the second set but was only able to win 13 points in the decider.
Third seed Suarez Navarro edged through, coming from behind to beat Romanian Alexandra Cadantu 2-6 6-4 6-4.
The Spaniard was broken three times in losing the opening set, but hit back in the second, making the crucial break in the 10th to level the match.
The decider was a back-and-forth affair, the duo each losing two service games early in the set, before once again Suarez Navarro broke in the 10th to seal victory.
There was another shock as eighth seed Tsvetana Pironkova crashed out in straight sets to Shahar Peer, 6-2 6-1.
The Israeli dominated her opponent, breaking her seven times in the match to complete a resounding victory in just one hour 15 minutes.
Fifth seed Klara Koukalova fared better, although she had to come from behind to see off Annika Beck 4-6 6-3 6-1.
After losing the first, the Czech came back strongly in the second, breaking her opponent three times to send the match into a final set.
Koukalova proved too strong for the German in the decider, stringing together six successive games from 1-0 down to book her quarter-final berth.
In a first-round clash, seventh seed Magdalena Rybarikova enjoyed a more straightforward passage, overcoming French qualifier Claire Feuerstein 6-2 7-5.
The Slovakian broke her opponent in her opening two service games and never looked back in the first set, before earning the crucial break in the 11th and holding out for victory.
Elsewhere, Croatian veteran Mirjana Lucic-Baroni got the better of Jovana Jaksic 6-3 6-4 to set up a last-16 meeting with Yvonne Meusburger.