Mauricio Pochettino's men were always in control against the team from Spain's fourth tier, producing a professional display despite the absence of Gaston Ramirez, who has been linked with a move to Inter.
Guly was the star of the show, netting an impressive treble, with Adam Lallana, Jason Puncheon, Morgan Schneiderlin, Jay Rodriguez and Luke Shaw also getting their names on the scoresheet.
Puncheon, Lallana, new signing Dejan Lovren and striker Rickie Lambert were all included from the start, and it took Southampton just 11 minutes to find the breakthrough, with captain Lallana slotting past Palamos keeper Juan Bayona after 11 minutes following good work from midfielder Lloyd Isgrove.
That lead was quickly doubled as Puncheon latched on to a cross-field ball from Danny Fox to power home.
Southampton continued to push forward at will, and they were rewarded again after 36 minutes as Schneiderlin converted from close range.
Pochettino made nine changes to his side at the half-time interval, with Victor Wanyama, Shaw and Rodriguez all thrown into the fray.
The mass alterations did not affect the flow of the game, though, and the visitors' lead was quickly extended as Rodriguez converted from the penalty spot on the hour mark.
A superb long-range strike from Guly made it 5-0 after 63 minutes, and the Brazilian was on target again two minutes later, heading in a far-post cross from Nathaniel Clyne.
Guly was a constant thorn in home side's defence, and he completed his hat-trick after 72 minutes, pouncing on another back-post delivery from Clyne before Shaw rounded off the rout with a deflected left-footed drive five minutes from the end.