The England striker, 23, came through United's famed youth system but joined the Gunners for £16m on deadline day.
United brought striker Radamel Falcao on a £6m loan deal from Monaco, their sixth signing of the transfer window.
Media playback is not supported on this device
"They have probably lost the way of Manchester United a little bit," Phelan told BBC Sport.
"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.
"What will happen in the future now, nobody knows but that thread has been broken now."
Former United midfielder David Beckham said he was sad to see Welbeck leave, but is hopeful promising players will continue to be developed under new manager Louis van Gaal.
"Manchester United have always bought players in but also have had a great scouting system which Sir Alex Ferguson put in quite a few years ago, where we did create home-grown talent, and that is something we have done over the years," said the former England captain.
Media playback is not supported on this device
"I think Louis van Gaal is a talented manager, he is well respected and with Manchester United this is a project that will hopefully last for a few years and be a successful one.
"Hopefully we have still got young players coming through."
For many years United had a strong core of home-grown youth players under former boss Ferguson, with the renowned 'Class  of 92' - which included Beckham, Paul Scholes and Phil and Gary Neville - coming through the ranks.
As well as Colombian Falcao, new boss Van Gaal has also brought in Angel Di Maria for a British-record £59.7m, midfielder Ander Herrera for £29m from Athletic Bilbao, left-back Luke Shaw in a £27m move from Southampton, the £16m Marcos Rojo from Sporting Lisbon and midfielder Daley Blind from Ajax for £13.8m.
Meanwhile, United youngsters Tom Lawrence, Nick Powell and Michael Keane left on Monday, either on loan or permanently.
However, defender Tyler Blackett, a youth academy product, has played in the Premier League this season, while defender Reece James and goalkeeper Ben Amos have been included in the matchday squad.
But Phelan, who was Ferguson's assistant between 2008 and 2013, believes the big-money arrivals at Old Trafford could suggest a change of philosophy at the club.
"There is always the start of something and maybe this is the start of a new way of doing things at Manchester United and maybe that is the way football is going," he added.
"Is it better to look at the instant rather than the future? It is a difficult one because youth is always the future, we all have to start somewhere and you just hope that product of youth can develop in the Premier League."
Phelan expects Arsenal to give Welbeck the chance to flourish after finding his chances limited at United.
Gunners boss Arsene Wenger already has a number of England internationals in his squad, with Jack Wilshere, Theo Walcott, Kieran Gibbs and Alex Oxlade-Chamberlain all first-team regulars when fit.
England squad member Calum Chambers was signed in the summer from Southampton and the defender has made three Premier League appearances this season.
"He is still only a young player and with young players you will get peaks and troughs but it is a case of Danny feeling more important in himself than he probably was over the last 12 months," added Phelan.
"He is in a position where he can play where he wants and I think that is why he chose Arsenal because they give him the opportunity to play in his best position, which is a striker."

Media playback is unsupported on your device
29 July 2015 Last updated at 02:23 BST
Now he finds himself with a death sentence after having been in prison for almost four years.
BBC News charts the rise and fall of Colonel Gaddafi's favourite son.
Video by: Mohamed Madi
Former Manchester United assistant manager Mike Phelan feels part of the club's "identity has been broken" with Danny Welbeck's departure to Arsenal.
[NEXT_CONCEPT]
Not too long ago, playboy Saif al-Islam Gaddafi was heir-apparent to an oil-rich dictatorship.