Rennie, who replaced the sacked Robertson three weeks ago, also named Neil Barnes, coach of New Zealand provincial side Taranaki, as his lone senior assistant coach, while handing his Kobe Steelers assistant Mike Blair the attack portfolio.
Forwards coach Jason Ryan is the sole Robertson assistant retained in the All Blacks setup, with Scott Hansen, Tamati Ellison and Bryn Evans jettisoned.
"Neil provides outstanding leadership and strategic support; Jase (Ryan) has proven himself as one of the best forwards coaches in the game," Rennie said in a New Zealand Rugby statement on Tuesday.
"Tana brings immense 'mana' and defensive insight and Mike will bring his innovative approach and attention to detail to our attack.
"I know this group will challenge and support both me and our players to bring the best out of us all."
Umaga, who played 74 tests for New Zealand, with 22 as captain, said he was thrilled to land the All Blacks job even if it meant sacrificing his personal life.
"I've been fortunate enough to actually captain this side and that was a big job," he told New Zealand media on Tuesday.
"I had to understand that, then also understand what comes with that. The sacrifices that not just yourself, but your family have to make.
"I'm fortunate that I've felt that pressure before and ... I know what I'm getting myself into."
Rennie also brought in Phil Healey as head of performance, having worked with the strength and conditioning trainer at the Waikato Chiefs, Glasgow Warriors and Kobe Steelers.
Rennie, the first New Zealand head coach with Pacific islander heritage, will see out Kobe's season in Japan before preparing the All Blacks for the inaugural Nations Championship tests in July against France, Italy and Ireland.
NZR said Ellison had been appointed Maori All Blacks head coach to replace Ross Filipo who left New Zealand last season for a coaching role in Japan.