Police were called to Iceland in Havelock Square shortly after 11:30pm on Tuesday, December 30, after the alarm was activated at the store.
Descriptions of two suspects were passed to officers, who searched the area and arrested a man in his 40s on suspicion of burglary.
He is in custody at Gablecross for questioning.
Police say that the other suspect remains outstanding and they are continuing their enquiries.
The town centre supermarket remained closed on Wednesday morning, with a sign stating “Sorry for the inconvenience! We are closed until further notice. Please do not move (the) cages! This is a crime scene, police (are) attending.”
Officers investigating the incident are appealing for anyone with information to contact them.
A Wiltshire Police spokesperson said: “A man has been arrested following a break-in at a Swindon supermarket.
 “Police were called to Iceland in Havelock Square shortly after 11.30pm last night (December 30) following an alarm activation.
 “Descriptions of the suspects were passed to officers and following area searches, a man in his 40s was arrested on suspicion of burglary a short time later.
 “He remains in custody at Gablecross for questioning.
 “One suspect remains outstanding and enquiries are ongoing.
 “Damage was caused to a glass door and a cabinet containing alcohol inside the supermarket.
 “Anyone who witnessed anyone acting suspiciously in the Havelock Square area between 11pm and midnight last night should get in touch.
 “Email 70059@wiltshire.police.uk and quote crime reference number 54250168698. Alternatively, call CrimeStoppers anonymously on 0800 555 111.”