The hosts lead Mexico 2-0 after Friday's singles at Brisbane's Pat Rafter Arena.
Rodionova beat the unranked Giuliana Olmos 3-6 6-3 6-1 in just over two hours before Saville thumped world No.562 Marcela Zacarias 6-1 6-0 in 51 minutes.
They need just one win in Saturday's reverse singles or doubles to book a spot in November's 12-team finals tournament as Australia, runner-up twice in the last five years, chase their first title since 1974.
On fire 🔥🔥— Billie Jean King Cup (@BJKCup) Daria Saville wraps up her match in 51 minutes!��#BJKCup | @Daria_gav | @TennisAustralia pic.twitter.com/oPfn6B8yVzApril 12, 2024
A recent world No.6 in doubles and a veteran of 24 Cup ties, Olmos is the only Mexican woman ever to sit inside the top 10 or win a WTA Tour event in singles or doubles.
She had the better of a nervy Rodionova, playing in her first tie since a 2016 debut, in a dominant first set and had break point at 3-3 in the second.
But the 34-year-old held serve, saving six of eight break points across the match, then swooped to break for the first time on her ninth attempt.
She won the next eight games, her only blemish coming at 5-0 in the third as she served for the match.
"Everyone knows, if you're looking by the rankings, we're supposed to win. But that's so dangerous" the Russian-born talent said.
"I just kept waiting, fighting and hoping she would go back down to the level she usually plays in singles.
"You can play tennis for 20 years ... then come out and play for something bigger than yourself and you're like, 'Which side do I hold my racquet?'.
"It took a few games to relax and remember it's still the same game, but mentally it's very different."
Victory - behind 25 Rodionova winners to Olmos's 13 - was sweet relief after a horror injury to Australia's No.2 Storm Hunter in practice on Thursday.
Storm Hunter was in the stands to watch Australia's progress, despite her Achilles injury. (Darren England/AAP PHOTOS)
The Queenslander ruptured her right Achilles tendon, the injury ruling her out of the Olympics and likely the remainder of this year's WTA Tour.
Saville, the former world No.20 who re-entered the top-100 earlier this week, replaced Hunter and made the most of the call-up to improve her Cup singles record to 5-7.
"I thought I played pretty well ... I thought I was pretty determined and that shows.
"Yesterday was a tough day, pretty emotional ... we're giving Stormy a lot of love."
Captain Sam Stosur is yet to announce who will play doubles for Australia, after year-ending world No.1 Hunter and Ellen Perez were initially selected.
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