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Kroos broke the deadlock with a superb 20-yarder into the top right-hand corner and hit the woodwork late on in a man of the match display.
The 24-year-old, whose contract expires in 2015, is yet to sign a new deal with the European champions, which has inevitably alerted United.
Moyes wants to bring in at least two central midfielders in the summer in an attempt to belatedly address the Reds' malaise in that area. In the last six years, Marouane Fellaini is the only midfielder United have bought.
Kroos' display reaffirmed his importance to Bayern, even though he is arguably their fifth-choice midfielder behind Bastian Schweinsteiger, Javier Martínez, Thiago Alcântara and Philipp Lahm, who switched from right-back to the middle in the second-half last night.
Moyes rejected the chance to sign Thiago in the summer and has scouted a number of top-drawer midfielders this season. The Scot has travelled across the continent to watch Atlético Madrid's Koke, Juventus duo Paul Pogba and Arturo Vidal and Paris Saint-Germain's Blaise Matuidi.
Last month, Moyes watched Kroos start at Borussia Mönchengladbach and was pictured in conversation with the Germany international's agent.