American Ginepri, who won the Indianapolis title in 2005 and reached the semi-finals last year, was upset by 83rd-ranked Michael Berrer of Germany 3-6 6-2 6-1.
Berrer, 27, who reached the quarterfinals at Hertogenbosch on grass last month, lost to countryman Nicolas Kiefer in the quarter-finals at Los Angeles last week.
"I'm full of confidence," Berrer told reporters. "For the moment I'm in good shape. I want to win a lot.
"Maybe today I wanted it a little bit more than he did."
Ginepri won the first set after breaking Berrer's serve in the eighth game but Berrer raced to a 4-1 lead in the second set after breaking serve in the second game, and then broke serve again in the eighth.
The German then took a 3-1 lead in the final set when Ginepri double-faulted twice in a row to go to 0-40 and netted a volley to lose serve in the fourth game.
Eighth-seeded Becker of Germany, the only other seed in action on Monday, was also a casualty as he fell to Frank Dancevic of Canada 6-4 6-3.
The 109th-ranked Dancevic broke Becker's serve three times in the match - in the third game of the first set and the third and ninth games of the second set.
Sam Querrey, the world number 90, ended a six-match losing streak which dated back to May with a 7-6 6-1 first-round win over Frenchman Julien Benneteau.
The 19-year-old, who blasted 19 aces, lost only four points on first serve in the match and reversed a run of form that has seen him win just once in nine matches.
"This was my first win in like seven weeks," Querrey said. "This was a very important win. Every time you win, it builds on your confidence.
"This was a big boost."
Russian teenager Evgeny Korolev made good on nine of 12 break points to beat American veteran Justin Gimelstob 6-2 6-4 and book a place in the second round against top seed Andy Roddick.