Danny Welbeck and Federico Macheda led the home attack, with Gabriel Obertan also making a start, while the likes of Michael Owen, Michael Carrick, Paul Scholes and Darren Fletcher all started on the bench.
United dominated the early minutes, with Darron Gibson to the fore and 13 minutes in Anderson's flag-kick was met by Nemanja Vidic, but he failed to direct his header on target.
However, an already quiet Old Trafford was silenced 20 minutes in as Rodrigo Tello picked up the ball all of 30 yards out on the right and his left-footer beat Ben Foster going away to his right, with the aid of a slight deflection.
Welbeck almost found an equaliser when visiting keeper Rustu Recber made a fine block following another fluent United move and the same player denied Gabriel Obertan with a similar save on the half-hour mark.
Four minutes before the half-time interval Besiktas passed up a superb chance to double their advantage, as a quickfire counter-attack ended with Michael Fink sidefooting against the outside of the post from 12 yards when he should have done better.
The second half got off to a slow start, with United struggling to find a way through a packed visitors' defence.
Just past the hour mark Obertan drove forward and his low, left-footer was tipped away for a corner by Rustu.
From that point on United rarely looked like getting back into the game, but in added-on time they carved out a couple of excellent openings.
First Rustu made a superb finger-tip save after Anderson dinked in a cross and Macheda flicked a header goalwards from five yards out.
From the resulting corner Rustu did even better to tip away a Wes Brown header and his efforts ensured that Besiktas held on to record a famous victory.