Becker finally got the better of the game American 7-6 (7-3), 6-7 (1-7), 7-6 (7-4) and moves on to face top seed Andy Roddick in Wednesday's second round.
Roddick has beaten Becker twice but the two have not played each other since 2007.
Ginepri led the third-set tiebreak 4-1 but Becker rallied to win six consecutive points.
In other play Tuesday, defending tournament champion Juan Martin Del Potro of Spain came back to beat Taiwan's Yen-Hsun Lu 4-6, 6-3, 6-2 and Germany's Tommy Haas, a semi-finalist at Wimbledon, topped Frank Dancevic 6-1, 4-6, 6-4.
Also advancing Tuesday was Juan Carlos Ferrero of Spain, a former No 1 ranked player, who beat Ecuador's Nicolas Lapentti 6-4, 6-7 (5-7), 6-0.
Copyright 2009 by United Press International.