United boss Sir Alex Ferguson will see this as two points dropped after they dictated the first half - but his side became unsettled by Everton's highly-combative approach in the second period.
In contrast, Everton counterpart David Moyes will be delighted at how they recovered from an early pounding to earn a draw, thanks to their £15m record signing.
United dominated the opening exchanges, although it took a superb block from Wes Brown to halt Louis Saha as he found himself in space on the edge of the area.
And a prolonged period of possession ended with United almost inevitably taking the lead after 22 minutes.
Interview: Man Utd assistant boss Mike Phelan
Giggs was the creator with a clever pass into the box, giving Fletcher the chance to take a vital touch away from Joleon Lescott and beat Howard with a composed finish.
Everton were in serious danger of being totally over-run, and keeper Howard was their saviour shortly after with three vital saves in the space of a minute.
He dived to his left to save from Cristiano Ronaldo, blocked Nemanja Vidic's header from the resulting corner and then produced an outstanding stop from a 20-yard drive from Giggs as United controlled the game with almost embarrassing ease.
Wayne Rooney was in search of his 100th career goal against his former club, but he missed an opportunity seven minutes before half-time when he saw a close-range effort scrambled wide by Joseph Yobo.
Everton had a half-chance a minute later following a mix-up between Vidic and Van der Sar, but Yakubu lifted his effort well off target.
Moyes' side needed to do something to at least to gain a foothold in the game, and they understandably adopted a more physical approach at the start of the second half.
Brown's mistake almost created an opening for Fellaini, and there there was a flashpoint when a challenge from Neville flattened Ronaldo.
Interview: Everton boss David Moyes
Ronaldo completed several rolls before Rio Ferdinand raced yards to get needlessly involved, the tussle ending in a yellow card for the Everton captain.
Everton's renewed vigour had shaken United's composure, and their spirits were lifted further when the giant figure of Fellaini drew them level after 63 minutes.
Neville lofted a cross into area and the Belgian, who had troubled United with his aerial prowess, rose to beat Van der Sar with a downward header.
Matters almost got worse for United a minute later when Ferdinand's poor back pass let in Yakubu, only for Van der Sar to divert his shot on to the post.
The setback upset Rooney, who was harshly booked for a collision with Mikel Arteta. He then foolishly gestured at his badge in front of the Everton fans to earn another rebuke from the referee - and was then swiftly removed, to be replaced by Nani.
Giggs almost created an opening for Ronaldo with six minutes left, but Joseph Yobo made a crucial interception.
Ronaldo, who had been given an uncomfortable afternoon, then dragged a highly-presentable chance wide from only 12 yards as United finally regained some momentum.
It was the final serious action of a pulsating encounter, that ended with Everton more satisfied with the result than United.
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