The pair have signed permanent deals until 2017, with the option for a further year, with the west London club as Rene Meulensteen added five new faces on deadline day.
As exclusively revealed by Goal, the Cottagers, whose manager, Rene Meulensteen, first joined United as a youth coach in 2001, agreed fees for the pair on Thursday before medicals were completed on Friday morning.
The double deal rounds off a busy deadline day for the west Londoners, who also brought in Lewis Holtby on loan, Konstantinos Mitroglou and John Heitinga.
Speaking to the club's official website, Tunnicliffe said: “I’ve known Rene since I was nine and there was talk when he took over that he was interested in me, so it’s just picked up steam this week really and I’m here now and glad to sign.
"Rene didn’t have to sell Fulham to me, for a start they’re in the Premier League and I want to play at the highest level. I know Rene and I know he’s a great coach so I’m just excited to work for him.
“Fulham’s a great club anyway – they’ve been in the Premier League for a long time now and I just can’t wait to get started.
“I want to play at the highest level I possibly can and, to me, that’s the Premier League because it’s the best league in the world. I want to play as many games as I can and improve as a player because I’m only young, and we’ll see where we go from there.”
Tunnicliffe had a recent loan spell at Ipswich, making 27 Championship appearances this season, while Cole scored twice in United's 3-2 victory over Tottenham in last season's Under-21 final.