Drakes, whose father Vasbert played for Warwickshire in the 2000s, will replace Hasan Ali as one of Bears' two overseas players, with the Pakistan seamer departing for international duty this weekend.
Drakes, 25, has been capped 12 times in white-ball internationals and enjoyed had a spell in county cricket with Yorkshire last summer.
"We're really pleased to welcome Dominic to the squad. Dominic loves the big stage and is comfortable taking the ball in the powerplay and high-pressure moments," Warwickshire's Pperformance director, Gavin Larson, said.
"As a left-armer, Dominic will add something different to our pace attack and we can't wait to see how he fares before the quarter-final."
Birmingham are top of the North Group with two games to play and assured of a spot in the Blast quarter-finals.
"I can't wait to link up with the squad ahead of what's an exciting time for the club," Drakes said. "The team have shown to be one of the strongest in the competition and I'm looking forward to adding to their initial success in the closing stages.
"I love playing in big crowds and having the opportunity to play in a quarterfinal, plus the chance to go one step further to Finals Day is exciting."