Ms Tully, who is chief customer officer for Australia's national carrier, will take over from Gareth Evans in November.
She will be replaced by Markus Svensson in her current role.
Jetstar apologised earlier this month after its fleet was hit by problems ranging from lightning and bird strikes to a missing spare part, disrupting the travel plans of thousands of passengers.
Qantas last month reported its third consecutive annual loss because of the coronavirus pandemic as border closures and travel uncertainty weighed on earnings.
"These appointments come at an important time for us," Qantas chief executive Alan Joyce said on Monday.
"The team is working incredibly hard to overcome challenges facing the whole industry as it gets back on its feet, and the data shows we're almost there.
"Managing this kind of executive renewal internally means we keep our momentum and can leverage a huge amount of corporate knowledge, including through the transition."
Qantas had announced in June that Mr Evans would step down as Jetstar chief by the end of the year.
Ms Tully joined Qantas in 2004 and has held senior operational, commercial and marketing roles including three years as a group executive.
"She's an outstanding leader and she'll be leading a very experienced senior team at Jetstar to keep building on the strengths of that business," Mr Joyce said.
Australia's national carrier posted a full-year underlying pre-tax loss of $1.86 billion, a figure described by Mr Joyce as "staggering".
The group's total losses throughout the pandemic have reached nearly $7b, although passenger demand has rebounded with the removal of travel restrictions.
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