Antonio Candreva got the only goal of the game midway through the second half for the Serie A club
Antonio Candreva got the only goal of the game midway through the second half for the Serie A club, who were forced to play out the final 13 minutes a man down after Marcelo Brozovic was sent off.
It was a cruel result for Southampton, who had been the better side and saw Jay Rodriguez have a goal disallowed early in the second half.
The result sees Saints remain on four points in Group K, two points behind Sparta Prague, who beat Hapoel Beer-Sheva 1-0 in Israel.
"It's very tough, to concede a goal with their only chance is frustrating," away captain Virgil van Dijk told BT Sport.
"That's what happens when you don't score your chances. We had so many opportunities to win the game, but now we need to move on."
For Inter, it was a first three points in the competition and the win against a team in good form in the Premier League should buy Dutch coach De Boer some time as the Nerazzurri struggle for results.
"I have faith in my players. We have lacked concentration in other games, but today we played with discipline for the entire match," De Boer said.
"I want to see us play with that malice all the time. It's fundamental."
Captain Mauro Icardi, forced to issue an apology to fans in midweek and promise to reprint his autobiography after including an allegedly distorted description of a 2015 altercation with the club's hardline 'ultra' supporters, started for the hosts.
Southampton boss Claude Puel made a host of changes to the team that beat Burnley at the weekend, including giving a first start at left-back to the young Sam McQueen.
It had the air of an historic evening for the English side, who were in the third tier when Inter won the Champions League in 2010.
An army of seven thousand travelling fans from England helped generate an atmosphere at a largely empty San Siro, but the Saints supporters were left frustrated by their team's inability to translate their dominance into goals.
Yuto Nagatomo had to clear off the line to prevent goalkeeper Samir Handanovic deflecting a McQueen cross into his own net just before the half-hour, and James Ward-Prowse fired narrowly over after being set up by the impressive Cuco Martina.
Rodriguez, sliding in at the back post, was then just unable to get a touch after Shane Long had flicked on a Martina cross, while all Inter offered in the first half was an Eder strike that flashed narrowly past.
The visitors replaced the injured Long with Charlie Austin just after half-time before Rodriguez had the ball in the net in the 51st minute.
After he had been prevented from getting a shot away by a great Joao Miranda tackle, he headed in from the subsequent corner but the Lithuanian referee's whistle had already blown for an apparent foul by the striker.
Inter went on to take the lead on 67 minutes as Davide Santon delivered a low ball from the left and Candreva came across Van Dijk to finish emphatically into the roof of the net past Fraser Forster.
Croatian midfielder Brozovic was dismissed for a second yellow for catching Pierre Hojbjerg, and Southampton laid siege to the home goal in the closing stages.
However, Van Dijk was denied by Handanovic at point-blank range and then saw a header cleared off the line.
Handanovic produced a great save from Austin before Ward-Prowse volleyed just past in the 95th minute.
Saints will hope to gain revenge when the teams meet again at St Mary's in two weeks.