Morgan's first goal of the season was enough to secure Leicester's fourth successive 1-0 win and came after 38 minutes when he out-jumped Jordy Clasie to power Christian Fuchs's cross past Fraser Forster.
The victory moved them on to 69 points with six matches remaining, seven more than Spurs who drew 1-1 with Liverpool on Saturday.
Leicester manager Claudio Ranieri told Sky Sports: "Clean sheet, three points, step by step. It's fantastic because we know very well Southampton came here with a lot of confidence.
"I said to the players, it's the chairman's birthday, he needs a special present from us. The skipper came back from a long journey with the national team, it's amazing.
"We are dreaming, if we fight we can do something and we must continue. It's important to stay very calm and maintain our feet on the ground."
Morgan said Leicester were determined to take advantage of Spurs' result at Anfield by beating Southampton in front of their own fans at the King Power Stadium.
