Saints will pay Luton in the region of £2.5million compensation with Jones taking charge of his first game at Liverpool on Saturday.
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Jones, who is in attendance at Southampton's Carabao Cup tie against Sheffield Wednesday, will be joined at St Mary's by Luton coaches Chris Cohen and Alan Sheehan.
The south coast club had been looking for a new boss after axing Ralph Hasenhuttl on Monday morning.
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While a 4-1 defeat to Newcastle at the weekend had spelled the end for the Austrian coach, Southampton moved quickly with the club sitting in 18th spot in the table.
While his time at the club ended with a 2-0 defeat against his old club Stoke City, there's no doubt that the Hatters will be gutted to lose a coach of the 49-year-old's pedigree.
A move to the Premier League was always going to be tough to turn down, however.
Jones has done a remarkable job at Kenilworth Road, taking the Hatters from League Two to second in League One in three years.
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The Welshman left to join Stoke in January 2019 but was sacked nine months later. Jones then went back to Luton in the summer of 2020, guiding them to the Championship play-offs last season.
Luton are currently ninth in the Championship with 29 points from 20 matches.