The Cottagers climbed out of the bottom three as goals from Hugo Rodallega, Lasse Vigen Christensen and Ross McCormack saw off Huddersfield 3-1.
And Symons, who was appointed as full-time manager following a spell as caretaker, said: “The top six is possible.
“It’s a big ask where we come from but my aim is to improve every week. If we can do that then who knows where it will take you.
“The league is wide open. For us to get up there we need to go on a good run which is a monumental task.
“But I was caretaker when we started the run at Crystal Palace. We snuck in for the last day for sixth spot and beat West Ham in the play-off final. It can be done but Palace was a long time ago. It’s not against the realms (of possibility).”
He added: “People have had a tough time for different reasons and Hugo is one of them. The response from everyone has been fantastic.
“I preach to them about trust, honesty and belief.
“They were three good goals. We have got some good players and some of the link-up play is a joy to watch at times.”
Fulham rode their luck early on but soon found their feet and took a 19th-minute lead when Rodallega slotted home.
Chances went begging for both sides, with Harry Bunn the main culprit for Huddersfield. But the Terriers deservedly drew level when Joel Lynch headed home Paul Dixon’s 75th minute free-kick.
Within two minutes Christensen chipped in a fine solo goal and McCormack’s long-range drive sealed the win with six minutes left.
Huddersfield have now slipped to successive away defeats and boss Chris Powell said: “On the balance of play, the way we started was very positive. The way we conceded the first goal having had a lot of play was disappointing.
“The response was good and after working so hard to get back in it we made the criminal mistake of falling asleep thinking you have done your job and conceded.
“There’s some good footballers in our squad. But if you don’t take chances then you are liable to concede. At this level you have to be clinical when on top.
“We are a side that can compete at this level and we have shown that. We have played two good sides in Derby and Fulham and come unstuck in the last week.
“They both have some players with Premier League quality and that’s what we aspire to. It was an open game and on balance it could have gone our way. We have two weeks to recover to get ourselves and our game-plan right.”