Southport arrived at the World of Smile Stadium having lost all of their previous five matches against Cheltenham Town and that record did not change on Saturday.
The Sandgrounders conceded three times in the second half as they suffered their second defeat of the season, leaving them without a goal after three matches.
Having shown encouraging signs to draw at Macclesfield Town in midweek, this was a harsh lesson for Southport, who had defended stoutly during the first half but struggled throughout the game in an attacking capacity.
Manager Paul Carden admitted it was a case of “men against boys”, but for all of Cheltenham’s possession in the first half, the best chance was created by Southport.
As Mike Phenix brought the ball forward he was stopped by Aaron Downes, but the loose ball broke into the path of Louis Almond, whose low strike was heading for the bottom corner until the finger-tips of goalkeeper Dillon Phillips intervened.
For Southport, however, that was the sum of their efforts and they failed to create another clear chance.
Port goalkeeper Andy Coughlin parried a 25-yard driver from Kyle Storer, while Jack Munns sent a first-time strike wide from Harry Pell’s cross.
The Sandgrounders had actually started the second half well, but within six minutes of the re-start they were behind.
Cheltenham had racked up 12 corners by that stage, and they eventually made one count, Munns’ delivery being converted at the far post by the head of Downes.
From then on, the home team were in cruise control and, after chances for Pell and Munns, Cheltenham added a second after more good play from Munns released Amari Morgan-Smith, who lifted a cheeky chip over goalkeeper Coughlin.
And Morgan-Smith claimed another with eight minutes remaining, nudging the ball home after Southport had failed to clear a Munns corner.
Southport: Andy Coughlin, Matt Challoner, Luke Foster, Josh Thompson, Will Beesley, Liam Nolan, Louis Almond, Charlie Joyce, Gary Jones, Courtney Cameron (Jamie Allen 57), Mike Phenix (Connor McCarthy 66).
Subs not used: Jon Viscosi, Adam Blakeman, Paul Rutherford.
Cheltenham Town: Dillon Phillips, Jack Bartham, Dan Parslow, Aaron Downes, George McLennan (James Bowen 82), Harry Pell, Kyle Storer, Jack Munns, Billy Waters (Asa Hall 66), Danny Wright (Joshua Cooke 83), Amari Morgan-Smith.
Subs not used: Calum Kitscha, Joe Hanks.
Attendance: 2,251 (107 away)
Referee: Robert Whitton