Jason Puncheon's first-half strike was enough to ensure Saints ended their week on the continent with a 100% win record.
Having already beaten Swiss side Étoile Carouge 4-1 on Wendesday, this win over the French Ligue 1 outfit will have left manager Adkins pleased with his side's fortunes during their time in Évian-les-Bains.
Adkins tinkered with the team which started Wednesday's win over Étoile Carouge, bringing Jay Rodriguez and Jack Stephens into his first-half line-up.
That meant Kelvin Davis was protected by a defence consisting of Danny Butterfield, Stephens, starting skipper Jos Hooiveld and Danny Fox.
With the gaffer sticking with the 4-3-2-1 formation used against Carouge, central midfield was again made up by Morgan Schneiderlin, James Ward-Prowse and Adam Lallana.
Puncheon took up a position on the right flank, with Rodriguez on the opposite side supporting out-and-out frontman Rickie Lambert.
The two sides were fairly equal during the opening exchanges, with only a Yannick Sagbo shot that deflected wide off Hooiveld and a close-range miss from miss by Fabrice Ehret to report from the first 10 minutes.
Saints, meanwhile, saw a Lambert free-kick sail over the ETG goal, while a Rodriguez shot from the edge of the area deflected just wide on the half-hour.
The visitors moved into the ascendancy during the final third of the opening period, with only a superb save by Bertrand Laquait preventing Rodriguez from heading in Ward-Prowse's miscued half-volley.
The resultant corner was cleared behind, before the next flag-kick saw Puncheon exchange passes with Schneiderlin before smacking a left-footed onto the crossbar and over.
Saints found their breakthrough ten minutes before the break as the pair combined again, with Puncheon retrieving Schneiderlin's short pass before advancing this time making no mistake as his low effort flew past Laquait to open the scoring.
HALF TIME: ÉVIAN TG 0-1 SAINTS
Adkins made eleven changes at the break, meaning Saints started the second period with a completely different team from that which finished the first.
That meant a debut for newly-signed goalkeeper Paulo Gazzaniga behind a defence consisting of Frazer Richardson, Fonte, Dan Seaborne and Dan Harding.
Midfield was made up by Calum Chambers, Dean Hammond and Richard Chaplow, while Luke Shaw and Guly acted as wingers just off Wednesday's hat-trick hero Billy Sharp.
Saints went close to doubling their lead just three minutes after the break as Shaw burst through the Évian half before squaring to Chaplow, but his controlled effort was tipped away from goal by Lacquait.
Much of the second half was played at a far slower pace than the first 45 minutes, with chances few-and-far between until the ball fell favourably for winger Daniel Wass just past the hour mark only for the Dane to put his half-volley just wide.
Youngster Shaw was withdrawn midway through the second period, leading to a debut for Nathaniel Clyne, who signed for the Club on Thursday.
The change almost inspired Saints to find a second as Chaplow dispossessed an opposition midfielder before running at goal and passing to Sharp, but the forward's left-footed shot was saved by Lacquait's legs.
Clyne, playing on the right wing, went close to having an immediate impact soon afterwards as he set Chaplow clear, but the midfielder curled wide of the far post in a promising position.
Les Roses poured forward in search of a leveller but generally failed to trouble Gazzaniga on his first run-out in Saints colours.
Instead it was the visitors who had the better chances to find the net, with Hammond watching a fierce shot go just the wrong side of the post with seven minutes remaining.
Saints' best chance of the half arrived in the 90th minute, as good play between Richardson and Clyne led to Chaplow cutting the ball back for Chambers, but his close-range shot hit an Évian man and rebounded to safety.
FULL TIME: ÉVIAN TG 0-1 SAINTS
Évian TG (4-4-2): Laquait; M'Baki, Partsvinya, Angoula, Ehret; Wass, Sorlin, Lacour, Dhaouadi; Sagbo, Bérigaud. Subs not used: Bugnet, Tie Bie, Rabiu, Dowbrie, Megdalia, Thomasson, M'Madi.
Saints (4-3-2-1): Davis (Gazzaniga 46); Butterfield (Richardson 46), Stephens (Fonte 46), Hooiveld (Seaborne 46), Fox (Harding 46); Ward-Prowse (Chambers 46), Schneiderlin (Hammond 46), Lallana (Chaplow 46); Puncheon (Guly 46), Rodriguez (Shaw 46 (Clyne 69)); Lambert (Sharp 46).
Goals: Puncheon (35)
Bookings: Hammond (72)