A single goal from defender Victor Lindelof separated the two sides, giving the Reds a welcome three points heading into November's international break.
Following a wonderfully observed Armistice Day silence, the Reds began the afternoon in a calm fashion, looking to use patient and persistent build-up to break down a defensive-minded Luton side.
Rasmus Hojlund and Alejandro Garnacho had the clearest chances in the first half, with the visitors managing few opportunities on the counter, but United fans had to wait until the second half for this dominance in possession to pay off. Luton were unable to convincingly clear a Bruno Fernandes corner in the 58th minute, allowing the ball to eventually drop to Lindelof inside the box and the defender duly rocketed it into the back of the net.