Long-range strikes from Abdoulaye Doucoure and Daryl Janmaat saw the Hornets pick up all three points from their visit to the south coast as Southampton drew a blank for the fourth time in their five outings across all competitions this season.
The Saints did not even have a meaningful shot on target until stoppage time of the second half, and Pellegrino admitted that his side will need to improve in front of goal.
"They were better in the first half especially with long balls and second balls. Physically, they were fresher than us. They created enough to score. We have to learn from today. In the second half, we were much better but we couldn't create," he told reporters.
"The game is 22 players to analyse not just 11. I will talk about my team and the other manager will talk about his. They had a lot of people behind the ball and it was difficult for us. It was tough. They were better than us in the first half.
"Southampton, for this reason (struggles in front of goal), changed the manager. If not and everything is perfect then maybe Mauricio would not be here. It's difficult to create chances and score goals but we have to keep going. We have some characteristics and quality and we can improve if we follow the dynamic of the last three games."
Southampton now sit 11th in the Premier League table with five points from their opening four games.