Current shortstop Cliff Pennington and the rest of the A's players found out by ballpark announcement late in the game.
The A's acquired Drew from the Arizona Diamondbacks on Monday in exchange for minor league infielder Sean Jamieson.
The trade, announced by Oakland before the seventh inning of Monday night's game with the Minnesota Twins, reunites Drew with former D-backs manager Bob Melvin -- in his first full season as A's skipper.
"It'll be refreshing and a new start," Drew said. "It's a good feeling to go over there and still be competing (for the postseason)."
Drew is expected to join Oakland in time to play the middle game of the series against the Twins on Tuesday. He recently returned after being sidelined for nearly a year with a fractured right ankle. He sat out 137 games in all.
The 29-year-old Drew is batting .193 with two home runs and 12 RBIs in 40 games since coming back June 26.
"Coming back, I've been hitting the ball well but haven't been finding any holes," he said.
Diamondbacks general manager Kevin Towers said last month it was "highly unlikely" the team would trade Drew this season, saying he hadn't received an offer for either Drew or Justin Upton that would improve the Arizona club.
A's GM Billy Beane must have changed his mind.
"We must be very persuasive," Oakland assistant GM David Forst said, noting the A's will see how the remainder of the season goes with Drew before making any decisions about his status with the club for 2013.
"It was more than likely he would have tested free agency so we probably would have only had his services for another six weeks," Towers said. "I have no doubt in my mind that he'll make the A's a much better ballclub."
Oakland, a surprise playoff contender this season with a cast of moving parts and new faces, had been looking for a consistent regular in the middle of the infield given the struggles and injuries of Pennington and Adam Rosales.
Pennington is hitting .198 with three home runs and 18 RBIs in 92 games. That includes batting 6 for 29 (.207) in 10 games since returning from the disabled list Aug. 7 after missing time with elbow tendinitis -- his first career DL stint.
"He plays the same position, so I guess we'll see what happens," Pennington said. "I was sitting there in the dugout and they announced it. I have no idea what those guys (in the front office) think. I'm going to try to pretend I do."
During the 2007 postseason playing for Melvin and the NL West champions, Drew batted .387 with two home runs and four RBIs in seven games, including .294 (5 for 17) against the Colorado Rockies in the NL championship series.
"I do know Stephen and he's a good player," Melvin said, preferring to wait until Tuesday to elaborate on the move. "He'll be a good fit here."
Drew is coming off the severe ankle injury and has a $10 million mutual option for next season. If the option is not picked up, Drew receives a $1.35 million buyout.
"I think he's probably recovered better than I thought," Towers said. "We have a lot of middle infielders and it's tough to spread the reps and at bats among three middle infielders."
The A's will make corresponding moves to their 40-man and 25-man rosters upon Drew's arrival. He is currently in an 0-for-15 stretch and is batting .069 (2 for 29) over his past nine games. Drew homered in consecutive games Aug. 5-7.
Second baseman Jemile Weeks said the announcement was "surprising to everybody."
"He's a good player, so if he comes along and jumps in and fits in and helps us win, we're all for it," Weeks said. "I guess they're trying to make a push, they're trying to improve what they feel they need to improve. They're trying to head in a certain direction. Obviously they have that vision in their heads and that's what they're trying to show right now."
The 23-year-old Jamieson, Oakland's 17th-round draft pick out of Canisius College in 2011, was batting .234 with 10 home runs, 49 RBIs and 25 stolen bases in 118 games for Single-A Burlington.
AP freelance writer Jonathan Dalton in Phoenix contributed to this story.