Koeman 'didn't recognise' Saints in first half
Relegation candidates Hull looked most likely to break the deadlock in the opening period, but came undone when Alex Bruce felled former colleague Shane Long in the box after 56 minutes.
James Ward-Prowse stepped up to net his first league goal, before Graziano Pelle scored domestically for the first time since December to seal maximum points.
"I think the second half was a comfortable win after the 2-0 but the first half was not good," Koeman told Southampton's official YouTube channel.
"It was one of the worst 45 minutes of the season at home and that was not good. I think we looked very nervous, that was strange. I didn't recognise my team on the pitch in the first 45 minutes.
"[The] second half was much better and finally it was a good win.
"I said in half-time the same words what I said here, that I didn't recognise the team. Why were we nervous? We don't have to be nervous, our opponents have to be nervous because they are fighting relegation."
The result lifted Southampton to fifth, with sixth-placed Liverpool to host Newcastle United on Monday.
"We have a good position in the table and we can enjoy and play our football," added Koeman. "It showed the character of the players and the team.
"We changed a little bit tactically in the midfield. I changed the positions in the midfield with Prowsey and Dusan Tadic and the team looked more comfortable."