Despite the Chile international reaching an agreement to join Manchester City, he made a shock move to Old Trafford towards the end of the January transfer window in a swap deal including Henrikh Mkhitaryan.
United manager Jose Mourinho hailed the deal as a "fantastic" on for his side, and Purslow agrees that it was the Red Devils who came out of it looking better.
"We will never quite know the sequence of events that led to that potential transaction winding away into a United purchase," the former Liverpool managing director told Sky Sports News.
"It has the look from the outside, I don't know this to be true, that Arsenal out of nowhere had their dreams come true.
"Rather than being in a unilateral discussion with City taking their best player nearly on a free they had a little auction going when Mourinho and Antonio Conte both spoke up in early January that actually getting Sanchez was a once in a generation opportunity to buy a top player for not much money down.
"I think Arsenal suddenly had a little auction going. I suspect when Sanchez's agent said to Manchester City's executive team: 'you might have to improve my deal up a bit because United are here' - that [would have] felt like a breach in trust.
"They thought they had a handshake deal at best. I bet they were very angry about that and I suspect they thought actually let's walk away from this deal leaving Manchester United to sweep him up.
"Brilliant outcome for United, pretty good outcome for Arsenal on balance but I suspect not a very happy camp at City around the way that deal was conducted."
Sanchez made his debut for United as they beat Yeovil in the FA Cup, playing a key role in their second goal, before playing the full 90 minutes in the 2-0 defeat to Tottenham in the Premier League on Wednesday.