The win was Inter’s first in Group K at the third attempt and the Serie A side more than rode their luck as Southampton dominated for long periods, created a string of chances and were desperately unfortunate not to earn at least a point.
Antonio Candreva rifled home what proved to be the only goal midway through the second half, but Inter were forced to play out the final 13 minutes with 10 men following the dismissal of Marcelo Brozovic and were grateful to goalkeeper Samir Handanovic for making several stunning saves as the visitors piled on the late pressure.
The result snapped a three-game losing streak for Inter to ease the pressure on De Boer, who is eyeing a top-two finish in the group.
“I must compliment the team,” he said. “We played with a lot of discipline.
“In the first half it was a bad game, but in the second half we did better and the goal from Candreva was fantastic. After the Brozovic red card we suffered, but I really want to give my compliments to the guys because we have held out to the end.
“In the last 10 minutes we struggled, but we have a fantastic goalkeeper.
“We have to continue like this and win every game if we are to make the top two in the group. If we keep winning, we will have the chance to go through.”
With Inter currently languishing in 11th place in Serie A having won only three of their eight games, there had been speculation in the Italian media that a defeat against Southampton would see De Boer sacked after less than three months in charge.
But the Dutchman played down suggestions that the players may have lifted their performance in a bid to keep him in the job.
“Did the boys play for me?” he said. “It is important that they played for Inter.
“I have faith in my team, in my players. In other matches they lacked concentration, but we played with discipline throughout the game. It is crucial for a team to always work together in order to achieve a victory like we did.”
Inter return to league action on Sunday with a trip to Atalanta.