Danish forward Sørensen, 19, won the Wor Jackie award, presented to the club's young player of the year, while County Durham-born left-back Gibson, 21, was presented with the Jack Hixon trophy for Tyneside's most promising youngster.
Sean Longstaff was last year's recipient of the Jack Hixon trophy - named after the scout who discovered Alan Shearer - while Cal Roberts, Freddie Woodman and Adam Armstrong are previous winners.
Woodman picked up last year's Wor Jackie award, following in the footsteps of Shola Ameobi, Steven Taylor, Andy Carroll, Aaron Hughes, Peter Ramage and Rolando Aarons.
Sørensen is currently on loan with Blackpool while Gibson is on loan at Accrington Stanley but both players attended the dinner at the Civic Centre Banqueting Suite in Newcastle, where a host of sporting personalities and dignitaries celebrated the very best of sport in the North East as well as launching Sport Newcastle's Tokyo 2020 appeal.
"It's a massive achievement," Sørensen told nufc.co.uk. "I feel very honoured to win this trophy. Many other very good players have won it before so I'm very proud. Since I've come to Newcastle, I've improved massively and the first half of this season showed what I was capable of."
Gibson has returned to action this term having being out of action for nearly two years after being diagnosed with ulcerative colitis, which led to him undergoing four operations and threatened not only his football career but, at one stage, his life.
He told nufc.co.uk: "I think the award is the final cherry on top of the cake. I'm really pleased to win the award and when you look at the past winners - the likes of Sean Longstaff - he's now got several Premier League appearances to his name so hopefully I can push on and maybe get to where he is.
"I'm trying to put all the negatives in the past now and I just want to play football. Hopefully I'll have a future at Newcastle."
Photograph: Ian Horrocks