New head coach Ronald Koeman witnessed his side score 10 goals in the victories over EHC Hoensbroek and Sporting Hasselt last week.
There was no repeat of the goal fest against the League One outfit at the County Ground on Monday, but the Dutchman will no doubt be happy to see his team collect another win and clean sheet.
Southampton failed to find the target in a lively first-half show, but it took just six second-half minutes for Hooiveld to stab past Swindon goalkeeper Wes Foderingham.
Graziano Pelle - who scored a brace on his first Southampton outing in the 6-0 rout of Hasselt last Thursday - was busy in the opening exchanges and showed good hold-up play before laying into the path of full-back Nathaniel Clyne, whose shot was parried by Foderingham.
The visitors again went close in the 16th minute when Matthew Targett's cross from the left found Lloyd Isgrove on the edge of the box but he curled his shot narrowly wide of the post.
After a goalless first half Southampton made the breakthrough in the 51st minute. Hooiveld got his foot to James Ward-Prowse's corner and poked past Foderingham.
The visitors almost had a second from their next attack when Emmanuel Mayuka's cross was flicked over the crossbar by the head of Ward-Prowse.
Swindon had created few chances but an unlikely equaliser nearly arrived in the closing stages when Connor Waldon rounded Paulo Gazzaniga, only for Calum Chambers to intervene with a well-timed tackle.
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