The 31-year-old has been a mainstay in Fulham's defence since joining the club as a relative unknown from FC Copenhagen five years ago.
Hangeland's current deal runs out at the end of the season and although manager Martin Jol remains confident the Norway international will commit his future to the club, Sidwell was quick to highlight his importance to the team.
"You want to keep all of your best players," the midfielder said. "His partnership with Hughesy (Aaron Hughes) has been instrumental to the club's success throughout the past few years.
"You want to keep your best players to progress as a football club in the Premier League.
"We lost (Danny) Murphy last year skipper-wise and Brede was a natural leader on the pitch when Murph was here.
"It was natural for him to get it and obviously his presence and performances week in, week out...you very rarely see him have a bad game.
"We obviously want to keep him. Full stop."
Hangeland returned to the Fulham starting line-up on Monday night after a three-match suspension and helped Jol's side earn a hard-fought 2-1 win against Newcastle.
"Brede is a stalwart for the club," Sidwell added. "I'm not sure how many games he has missed in his time here.
"Obviously, when he doesn't play he is sorely missed so just having him around the place lifts the players. Having him on the pitch as well, with him being our skipper, gives everyone a buzz.
"Defending from the back he pushes up and I thought he was brilliant last night."
Monday's win provided a timely boost for a Fulham side that came into the match without a win since October 20.
Hugo Rodallega's 63rd-minute header proved decisive in ending a seven-match winless run after Hatem Ben Arfa had equalised nine minutes earlier.
Sidwell opened the scoring in the first half with a deflected 19th-minute strike, ending a 317-minute wait for a Fulham goal.
"I can't put my finger on it," the midfielder said of the drought. "We were working hard, creating openings and obviously when you have somebody like Berba (Dimitar Berbatov) in the team, he is always going to create chances.
"We've had that in previous games but just not finished them and we have been letting in goals in Brede's absence so it just all piled on top.
"But now we have this win, we've got two goals under our belt and can kick on. We've got another massive game against QPR on Saturday."
Fulham may be back to winning ways but Newcastle's struggles continue.
The Magpies lie just five points off the relegation zone and summer signing Vurnon Anita knows things need to improve quickly.
"It is a frustrating run at the moment because we won last week and we wanted to get our first away win," the Netherlands international said.
"It didn't happen but we must keep on having faith, working hard to get the wins.
"Yesterday they were sharper than us at set plays but we must work on that, improve ourselves and go onto the next game.
"We have got the fans behind us so it will help us and I hope we can do our best on Saturday."