Rahane, 27, is first cricketer from Mumbai in 15 years to be handed the reins of the team since Tendulkar led the team in 2000.
The selectors announced a second-string team for the tour, resting most of the senior players such as MS Dhoni and Virat Kohli.
"I feel very happy for Ajinkya. He is a sincere and hard working cricketer. I have always been impressed with his commitment and dedication. I am sure he will give his best and give us a reason to be proud of him. I wish him all the best," Tendulkar told PTI.
Rahane's elevation has come days after he was dropped from the playing XI for the last two One-dayers in the recent Bangaldesh ODI series. Regular skipper Dhoni had justified Rahane's exclusion, saying he struggles to rotate strike on slow pitches.
India are scheduled to play three ODIs in Harare (July 10, 12 and 14) followed by a couple of Twenty20 Internationals (July 17 and 19).