Shaw's transfer, while expensive by any reasonable metric, is one that strengthens United's squad in a key area of weakness.
Patrice Evra has been an outstanding servant of the club but is reaching the end of his top-level playing career. Alexander Buttner was clearly not seen as a suitable long-term replacement and has been sold to Dynamo Moscow.
Shaw, still just 18 years old (he turns 19 on July 12) has the potential to be United's first-choice left-back for more than a decade.
His quality is clear to see. Playing regularly in the Premier League at 18 is a remarkable achievement in itself. To do so in as impressive a manner as Shaw managed, in the 2013/14 season with Southampton, marks out truly special potential.
Given the fee United have paid for his services, he had better be special. Writing in his blog for Paddy Power, Paul Scholes said: "For that money I want a centre-forward who’s going to score 30 goals a season."
With respect to one of the finest footballers I have had the privilege to see, Scholes is misguided here. United have an urgent need to strengthen at left-back, and Shaw is one of the best options available.
The money is better invested where it is needed than spent on a player who will "score 30 goals a season." United have more than one of those already.
Back in 2010, Sir Alex Ferguson famously said of the January transfer window, per Sky Sports: "I just don't see that player who can make a difference for us in terms of value or ability." In Shaw's case, the "value" may be questionable, but the "ability" is not.
Almost any player bought for a price tag such as Shaw's represents some kind of risk in terms of "value." However, the price begins to appear more reasonable when mitigating factors are taken into consideration.
Firstly, Shaw is a proven quantity in the Premier League already. There are very few 18-year-olds about whom that can be said. Shaw started 35 of Southampton’s league fixtures last season and maintained a good level of form throughout.
Premier League experience comes at a premium.
Secondly, United cannot offer Champions League football this season. This may be less critical in the case of Shaw than it would be to a player already at a Champions League club, but it is still a reason United's offer may have to be more financially attractive than a competitor's.
Thirdly, there were indeed competitors for Shaw's signature, if rumours are to be believed. On 10 June, Metro suggested that United's financial package was designed to "scare off" competition from Chelsea for Shaw's signature. If that was, in fact, the plan, then it has clearly worked.
If Shaw helps United's immediate return to the Champions League and goes on to be part of a team that is once again challenging for major trophies, then the price will certainly have been one worth paying. Given his quality, it seems likely that he will.
In previous Bleacher Report articles, I have pointed out the favourable comparison between Shaw and Evra and discussed the unique opportunity afforded to Shaw by having him arrive at the club while Evra is still in situ.
That has now become a reality; Shaw can begin to work with Evra on becoming a "Manchester United player" on and off the pitch.
In the space of a week, two very fine players have arrived at United in positions where help was most needed. Herrera arrives as the tackling, ball-playing, dynamic midfielder United have been crying out for.
Shaw arrives as the natural heir to Evra.
Between those signings and Louis van Gaal's masterful tactical management of the group stages of the World Cup, next season is starting to look a lot more promising.