Southampton became the last team in England's Football League to keep a clean sheet this season, and it was their first since a home win over Arsenal in April.
Hasenhuttl, whose side climbed four places to 14th, said it was an important win and that the players had delivered a special performance.
"We absolutely deserved to win today," he added. "From a team that from the first moment was showing we can do better and this was important to see.
“All the things we asked for from the players today they delivered in a perfect way - everybody in the starting 11 and the guys coming on the pitch - and this is how it should be every game, but it’s not that easy."
Hasenhuttl also said keeping Bournemouth out was great for goalkeeper Gavin Bazunu.
"His first clean sheet in the Premier League and he had to wait a long time, but sometimes you have to wait in your life to come into the right water.”
