The teenage midfielder fired his first senior goal as Chelsea eased past Scunthorpe 2-0 at Stamford Bridge on Sunday, booking their place in the fourth round.
Diego Costa's fourth goal in three games set the Blues on course before Loftus-Cheek sealed the win in the second half.
Guus Hiddink guided Chelsea to FA Cup glory in his first short-term Stamford Bridge stint in 2009 and now Loftus-Cheek believes the Blues must throw everything at helping the Dutchman repeat that triumph.
"Even if we were doing really well in the Premier League it's still a competition we'd want to win," Loftus-Cheek told Chelsea TV.
"But this season I think we should really want to win it.
"The Premier League hasn't gone that well, so I think this is a competition we've got to win."
Interim boss Hiddink admitted Loftus-Cheek could be called into more regular action in the weeks and months ahead after impressing in training.
Chelsea's 19-year-old academy product joked he would play in defence to get onto the field, before thanking the club's supporters for their strong backing.
"Right now I'm happy to play anywhere - right-back, left-back, wherever!" said Loftus-Cheek, when asked his preferred midfield role.
"I just want to get on the pitch, get some minutes and enjoy my football.
"The fans have been so supportive of me, right through from when I joined the first team.
"I feel a bond with the fans and it's a really good feeling to have them behind me."
Defender Gary Cahill echoed Loftus-Cheek's sentiments in calling on Chelsea to focus on cup glory this year, to atone for the horrendous Premier League form that cost Jose Mourinho his job.
"I think the FA Cup and the Champions League can be two massive competitions for us, and two that we want to go as far as we possibly can in," Cahill told Chelsea TV.
"Every game's important, but the FA Cup is definitely important for us this year."
Scunthorpe boss Mark Robins hailed his team for finding their feet and refusing to be overawed taking on the reigning Premier League champions.
Robins challenged his players to harness the confidence to kick-start a climb up the League One rankings.
"Moving forward we can take a lot of positives for the second half of the season," said Robins.
"You've got a feeling they are just toying with you and at any one moment they can open you up, but for the majority they didn't.
"We did pretty well, and it's a great experience for everybody to draw on."
Source : PA