The A's claimed Drew off waivers on Saturday, making Monday the final day the two clubs could finalize a deal, according to waivers rules.
Oakland had pushed to acquire the 29-year-old at the non-waiver Trade Deadline on July 31, but were unsuccessful because of injuries on the D-backs' roster.
"We had discussions with them but Willie [Bloomquist] had the bad back, so the timing wasn't right," Towers said. "We would've been very shorthanded. We wanted to get a better idea where we were at. We came close. It didn't happen, but we were able to make it happen through waivers now."
For Drew, Monday marked the end of his six-year tenure with the D-backs. The club drafted him 15th overall in 2004 out of Florida State.
"I was a little shocked but, first and foremost, I want to say thanks to those who supported me and to the organization," Drew said. "It's been a lot of fun. It's part of business. It's going to be good on both sides."
The shortstop said he was unaware that he was claimed before finding out about the trade after Monday's loss to the Marlins. Drew was 40 games into his season after coming back from a devastating ankle injury he suffered in July 2011 that caused him to miss nearly a calendar year.
In 135 at-bats this year, Drew batted .193 with 12 RBIs.
"It's kind of crazy, coming back and trying to get into a groove after being out so long," he said. "But my ankle is doing well, it really is. I've been pleased with it. That type of injury, you never know, but it's feeling good."
A move to Oakland will mean Drew will be reunited with his former manager, current A's skipper Bob Melvin.
"I'm looking forward to going back to an ex-coach. It'll be refreshing and a new start," Drew said. "It's a good feeling to go over there and still be competing."
With Drew gone, the D-backs will turn to rookie Jake Elmore and veteran Bloomquist, who is currently on the disabled list but is expected to return Friday, to carry the load up the middle.
"Elmore has played pretty good since he's been up here. He's played pretty comfortably and he handles the bat well," D-backs manager Kirk Gibson said. "We think we're getting Willie back, too, so we'll play them. We don't want to push Willie too hard, though."
As for Jamieson, the D-backs are getting a 23-year-old who hit .234 with 25 doubles, 10 home runs, 49 RBIs and 25 stolen bases in 118 games with Class A Burlington this year. He was selected in the 17th round of the 2011 Draft.
"We acquired a young shortstop, a Canadian kid who has speed and power for the position," Towers said. "He's probably a ways away, but he's somebody we think is a legitimate middle-infield prospect, which is something we've lacked in our system."
Jamieson will report to Class A South Bend for the D-backs.