The German pulled off a 7-5, 3-6, 7-6(7) comeback to notch the opening-day victory.
Brown rallied from being down 3-5 in the third set, refusing to let the Russian close him out. He saved four break points at 5-5 and later ousted Davydenko on his fourth match point in the tie-breaker.
Brown fired 13 aces and won 77 percent of his first serve points to put away the Russian in 2 hours, 4 minutes.
Also Monday, Spain's Guillermo Garcia-Lopez cruised past countryman Ruben Ramirez Hidalgo 6-2, 6-3 in a 90-minute match, and Belgium's Steve Darcis bested Albert Montanes of Spain 6-2, 6-4.