Gennaro Gattuso named a strong side that, for most of the first hour, looked incapable of breaking down theoretically hugely inferior opposition, and in the end their winning goal proved highly fortunate.
Milan were especially poor in the first half, hardly causing their modest opposition any problems at all at the back, with Dudelange arguably crafting the better opportunities.
But Gattuso's men stepped things up a notch after the break, finding the net - with a little help of a deflection - through Higuain in the 59th minute and that proved to be enough to open their Group F account.
Despite Milan's superior quality on paper, Dudelange actually crafted the better chances in the first half.
David Turpel fired over from Kevin Malget's cut-back in the 32nd minute, before Dominik Stolz followed suit from a tight angle after being released into the left side of the box soon after.
Milan eventually went close just before the break, but Mattia Caldara's header went well over despite being unmarked at a corner.
The visitors began the second half with renewed purpose and quickly went close - Joe Frising denying Higuain, who had met Diego Laxalt's cut-back.
But Higuain got lucky just before the hour, receiving Samu Castillejo's pass on the edge of the area and finding the net via a hefty deflection.
A Castillejo cross shortly after found Fabio Borini, who hit the post from close range, but it mattered little as Milan comfortably held on.