The Saints had seven shots on target during the fixture at St Mary's, but could not make the most of their dominance as Swansea returned to Wales with a point following the goalless draw.
Pellegrino has said that he was pleased with his team's performance on the afternoon, but the 45-year-old was frustrated with the failure to collect maximum points on the opening weekend of the campaign.
"We did everything to win the game but we couldn't. We controlled the ball, created chances but this is football - you have to get goals and we couldn't do so. As a manager, we can work on how to create chances but sometimes it is really difficult to score - we had 29 shots," Pellegrino told BBC Sport.
"Swansea is a good team, they were ready to attack us but we were well balanced and that was the key to controlling the game in the second half. For us it is best to keep going, keep training and we have another game on Saturday. I think we have quality and spirit in the team."
Pellegrino took the Southampton job earlier this summer after Claude Puel stepped down from the position.