The game was goalless aftre an hour before three strikes in 7 mins saw Saints go 3 - 0 up.
David Connolly (63, 68) and Michail Antonia (70) looked to have given the visitors safe passge to round 2 but a Darrly Duffy strike in the 73rd minute and an injury time penalty from Jeff Hughes made the result a lot tighter than Southampton fans would have liked.
The draw for round 2 takes place on Sunday.