The former Valencia star was given another chance on Wednesday night during the Copa del Rey clash with Hercules.
He made his fifth start of the season following his 30million euros move in the summer but the 23-year-old was unable to help his side grab a victory, and the game eventually finished 1-1.
Alcacer’s best chance came via a deflected free-kick and the Spain international is adamant his confidence has not been affected.
He told reporters: “There are a lot of people that want me to get the first [goal] and then kick on and I’m the first person who wants to score. I’m not going to give up just because the ball has not gone in yet. I have to keep a cool head and keep trying.
“It is not an overwhelming situation. I have to look at what my manager and the club are saying. I have to isolate myself from what the journalists are saying because they say good and bad things, and things can get in your head which aren’t true either way.
“I am getting chances to play and to score, but luck is not by my side. Some team-mates are even more keen for me to score my first goal than I am, so I hope [my luck] changes soon.”
Southampton have announced plans for a two-week tour of China as part of their pre-season preparations for the 2018-19 season.