First-half goals from Gareth Bale and Clint Dempsey proved the difference as Southampton only had a Jay Rodriguez goal to show for an impressive second-half display.
"The second half was what we wanted in the first half," Adkins told Sky Sports. "Credit to Spurs, they slowed the game right down.
"We had the sucker punch of the two goals we conceded, which were poor ones from our point of view. In the second half we had them on the rack but ran out of time.
"We totally dominated the second half and pinned Tottenham back. It was more of a cup tie and we had a high energy."
Adkins also praised Rodriguez, who scored his first Premier League goal for Southampton since his big-money move from Burnley in the summer.
The Saints boss added: "He works ever so hard and I think he's a special talent. He's got a great future in the game.
"His movement is exceptional. He got the goal he deserved today and had one or two other opportunities as well that didn't go into the back of the net."
Another signing Emmanuel Mayuka caught the eye and Adkins said of the substitute: "Mayuka coming on the pitch gave us a big impetus.
"He's a young man who has come in and wants to play down the middle. We introduced him down the right-hand side and he made the impact required, and did that very well."