Greece bowed out of the competition on penalties to Costa Rica on Sunday after the last 16 tie was taken to extra time courtesy of Sokratis Papastathopoulos' late equaliser.
Theofanis Gekas was the only man not to score his penalty as Michael Umana struck the decisive spot-kick to end coach Fernando Santos' Greek reign with a defeat.
The Hellenic Football Federation (EPO) are now tasked with finding a replacement for Santos, who took charge in 2010 and guided Greece to the quarter-finals of Euro 2012.
Sarris believes the right candidate is someone who will carry on the same path as Santos and his predecessor Otto Rehhagel, the coach during Greece's successful Euro 2004 campaign.
"The kids are disappointed about the way we were eliminated," Sarris told the EPO's official website.
"We must immediately begin preparing for the next Euros, which is very important for the continuity of this group.
"The work of Fernando Santos (over) these four years is invaluable. It is a continuity of the work of Otto Rehhagel.
"Mr Santos has further improved the infrastructure of the national teams (and) we all believe that the next coach will continue on the path of success that has defined this team."
