The prolific Colombian striker, one of the stars of the world game, joined United in a shock season-long deal from Monaco on deadline day.
His arrival capped an extraordinary summer for United in which they brought in the vastly experienced Van Gaal as manager and spent more than £150million on players.
Falcao, who was heavily linked with Real Madrid last week, is now looking forward to immersing himself in Dutch boss Van Gaal's philosophy.
The 28-year-old, whose £6million loan could be made permanent next summer, told MUTV: "I am very excited for that. Van Gaal is a very good trainer with a lot of experience.
"I want to do the best for the team, for him and for the supporters."
Falcao will join British record signing Angel di Maria, Wayne Rooney, Robin van Persie and Juan Mata in a formidable attacking unit.
Former United defender Rio Ferdinand, who left Old Trafford in the summer, is impressed by the recruitment.
Ferdinand, now at QPR, tweeted: "@ManUtd have gone huge! Falcao & di maria & the early signings...i hope they all fit in & gel. Need carrick fit too. The Firepower is nuts."
As Falcao arrived, United allowed homegrown striker Danny Welbeck to move to Arsenal for a fee thought to be around £16million.
Former United defender and assistant manager Mike Phelan believes that is a shame and signifies a major strategy change at the club.
Phelan told BBC Sport: "They have probably lost the way of Manchester United a little bit.
"Now, rather than produce, it may be the case where they are buying in.
"Someone like a Danny Welbeck has been part of United's identity and that has been broken."
Manchester City were linked with a move for Falcao early on deadline day but that never materialised.
Instead there was a high-profile attacking departure as striker Alvaro Negredo joined Valencia in a loan deal that will be made permanent next summer.
Negredo, 29, impressed in the first half of his single season at City but lost form after suffering a shoulder injury in January and is now out with a broken metatarsal.
Negredo posted a farewell message on Twitter reading: "About a year ago Manchester City, a club that for always will remain in my heart, gave me the amazing opportunity to play in England.
"For me winning the Premier League, the Capital One Cup and scoring 23 goals in a season is a dream come true... It has been one of the best years of my football career."
City captain Vincent Kompany tweeted: "Only one season at #MCFC but what a season! Wishing @_AlvaroNegredo_ a.k.a. The Beast, a successful return to @LaLiga and a speedy recovery!"
QPR had some deadline-day success as manager Harry Redknapp signed two players familiar to him in Niko Kranjcar, on loan, and Sandro, for £6million.
He also wanted a third in Jermain Defoe but Toronto were unwilling to release the former Tottenham striker.
It may not be the end of the matter though, with Redknapp telling talkSPORT: "There is every chance he will come in January.
"I'm a big fan of Jermain's. Almost certainly, I think we will (go back in for him). I've signed him so many times. I've taken him everywhere."
Southampton paid £10million for Red Bull Salzburg's Senegal forward Saido Mane, an exciting player who scored 45 goals in 87 games for the Austrian side.
Saints boss Ronald Koeman, who also completed a loan deal for Belgium defender Toby Alderweireld from Atletico Madrid, believes he joins the club at the right stage of his career.
Koeman told the club's YouTube channel: "It is a step forward to play in Southampton and the Premier League.
"Maybe for other teams, bigger teams, it is too fast for him.
"If he plays and shows his quality in some seasons he will get the possibility to and move to a bigger one.
"It is a good decision. Sometimes players need to be more patient."
Hull had a busy day, adding four players to the squad in Abel Hernandez - for a club record fee of around £10million - Mohamed Diame, Gaston Ramirez and Hatem Ben Arfa.
Manager Steve Bruce told Sky Sports News HQ: "On paper there are some really, really good signings we've made. We got what we wanted."