In the striker's two previous campaigns with the club, Albion suffered Championship play-off heartbreak against Crystal Palace and Derby.
The 31-year-old rejoined the club on loan from Leicester yesterday and says it does not matter who scores the goals, so long as Albion stay in the Premier League.
He said: "I hope that we can finish happily at the end.
"The last two seasons were sad because we fought, we tried to give everything and to do everything but it wasn’t enough. So I hope this time we can change things and the end is happy and we gain the objective, which is to stay in the Premier League.
"First of all because we are here but secondly because this club deserves to be here a long time. I think Brighton as a club improved a lot in the last five, six, seven years. They continue to improve all the time and so it is important for the club to maintain this status for a long time and I hope they do."
Ulloa, a fans' favourite after scoring 26 goals in 58 appearances in his first spell with Brighton, is not setting a goal target.
He said: "I’ll try not just to score as many goals as I can but give my best in training and in the games too. I’m here to try to help and at the end of the season we’ll see.
"Now is the dangerous position and we know it’s so difficult to stay in the Premier League and every game is difficult but we have to try to win the next game and stay together as a group. That’s most important thing for me, what I want to do.
"If five (goals) is enough, five, if ten is enough, ten. I want to help, don’t worry who (scores), the names are not important. More important is that we try to do this because that is the better thing for the club."
The man from Argentina said he would not have returned to Brighton if he did not think they could stay in the Premier League.
He netted 20 times in 102 matches for Leicester, becoming a Premier League champion during his time with the Foxes, but only after they won seven of their final nine matches the previous season to stay in the top flight.
He said: "I think the year winning the title started the year before when we fought together as a group for one object. It was safety. That was the moment we started to win the title.
"The next year was different but that was more important. The group has to start strong.
"That is more difficult in football. Every changing room is different and the players are different so when you fight together it helps everyone. Everything starts the same way and you can reach the objective. Okay, afterwards the players improved, but the other (team spirit) is more important, for me, than the tactics or what happens on the pitch.
"If I didn’t believe that (staying up), I’m not here. This group plays well and they have character. And character to play, not just defend the box all the time.
"They try in the Premier League to fight and to play but with a philosophy and that is very important, to continue what they did in the Championship. Try to play the same football. They create chances, they play football. Okay, you can lose but you play one way, you don’t change every time. I believe Brighton can do this."
Ulloa is looking forward to continuing his strong bond with the Brighton fans and added: "It’s special because I played just one and a half years here, not five or ten, but it was very intense. So I hope to keep that feeling but I know I have to do my job.
"I have to show them that I am still good and try to help the team. After that we’ll see what happens but I hope we can keep our relationship."
Ulloa is under contract at Leicester until the summer of 2019, after signing a new deal with the Foxes in August. Asked about his future following the loan spell with Brighton, he said: "I have to do a good job and after that if it’s okay we’ll make a decision. But I have to give my best in the next three and a half months. I just want to help the team."