Under the caretaker management of Symons, the Whites recorded just their second win of the season by narrowly beating Rovers 2-1 at Craven Cottage on Tuesday.
Symons has made no secret of his desire to permanently succeed Felix Magath, who was sacked last week with the club bottom of the Championship, and is optimistic that this result will help persuade the Fulham board that he is the right person for the vacancy.
"There will be a thorough search for the right manager and my challenge is to make sure that right person is me," he told reporters.
"Results are vital for my chances of getting the job. We have to build confidence - this is stepping stone and a good starting point."
Symons made over 100 appearances for Fulham as a player between 1998 and 2001 and has been a member of the club's coaching staff since 2008.