Gus Poyet's side climbed out of the Premier League relegation zone thanks to a comprehensive 4-0 victory against fellow strugglers Cardiff at the Stadium of Light on Sunday.
|1/4||Sunderland are 1/4 with bet365 to avoid relegation this season|
"We cannot assume that we have achieved anything. Nothing has been done yet – we have just started the job," Mannone told reporters.
"Before the Cardiff game, we had four games to go, and we talked about how they were four finals for us. We've only won one.
"There are still three more to go and they are every bit as important, if not even more. We have only won one battle. To win the war, you need to succeed in another few fights."
Mannone was also full of praise for in-form teammate Connor Wickham, who bagged a brace against Cardiff.
Wickham had gone 48 Premier League games without a goal for Sunderland but the English striker has scored five goals in three games to lead the team's rise up the table.
"He has all the talent in the world. Before, it was tough for him. But the work pays off," Mannone said.
"He went to the Championship, scored a lot of goals, and came back. That's a good way to be."
Sunderland face Manchester United at Old Trafford this week before ending their season with consecutive home fixtures against West Bromwich Albion and Swansea.