Sanchez made his debut in Beckham's old number seven shirt at Yeovil Town on Friday, helping set up two goals in a 4-0 FA Cup win.
"I still can't believe it's happened, to be honest," Beckham told Sky Sports. "When he's been part of a club as big as Arsenal and then he moves a few hours away up north to the biggest club, it's astonishing to see.
"We all wish him luck. I'm a Manchester United fan so I am excited about having someone of his talent wearing a Manchester United shirt - so it's exciting times for us."
Beckham also offered his support to Mourinho, with the Portuguese having extended his contract at Old Trafford just last week.
He added: "Of course he's the right fit for Manchester United. We're talking about Jose Mourinho. He's one of the best managers in the game.
"Manchester United has had success over the last 25 years because we have had stability that has run throughout our club.
"Sir Alex Ferguson - the amount of time he was manager of the team - that's why we were successful because we had stability that ran through the club and that's what we need going forward.
"Everyone is going to have their own opinion of [Mourinho] as a manager, him as a person, him as an individual. But is he the right person to lead Manchester United? Yes, of course he is."