The Frenchman edged the opening set at the ATP 250 grass-court event and, although Tomic fought back, the top seed progressed 6-4 3-6 6-3.
Gasquet won three games at the start of the final set, surviving five break points in the process, to give him the advantage before sealing victory on his third match point.
Defending champion and third seed Feliciano Lopez made sure of his place in the last eight with a 7-6 (12-10) 7-5 win over Tobias Kamke.
After a closely contested opening set, Kamke squandered two set points in the second before being eliminated.
Two fellow seeds did not have enjoyable days in Eastbourne as both Gilles Simon and Guillermo Garcia-Lopez suffered unexpected defeats.
Simon fell to a 6-3 6-4 reverse to French compatriot Julien Benneteau, and fourth seed Garcia-Lopez was beaten 6-2 7-5 by Denis Istomin.
Martin Klizan will meet Gasquet after his straight-sets win over Donald Young, and Istomin will face Edouard Roger-Vasselin after the Frenchman recorded a 6-2 7-6 (7-4) win over Jarkko Nieminen.
The final two spaces in the quarter-finals were taken up by Jeremy Chardy and Sam Querrey, who benefited from Victor Estrella Burgos' retirement.