Saints netted three times before the interval, before surging towards the finishing post with a further five efforts after the break.
Club captain Adam Lallana opened the floodgates with just eight minutes on the clock, and Jason Puncheon doubled the visitors' lead six minutes later.
It took until the 36th minute for Southampton to break through for a third time, with Morgan Schneiderlin getting in on the act.
Jay Rodriguez calmly slotted home from the penalty spot on the hour mark, before Guly took centre stage.
The Brazilian forward netted three times in the space of nine second half minutes - with Palamos carved open on 63, 65 and 72 minutes.
It was left to Luke Shaw to put the final nail in the hosts' coffin, with the teenage full-back completing the scoring five minutes from time.