The 21-year-old batting allrounder comes into the XI at the expense of front-line spinner Todd Murphy, who took middling match figures of 1-85 in last week's win.
A recent debutant in One-Day Internationals and Twenty20 Internationals, Connolly has played only four first-class matches and is yet to take a wicket at that level.
But on a pitch expected to turn from day one, he will be a second left-arm off-spin option next to Matt Kuhnemann - the leading wicket-taker in the series opener.
Former Test star Simon Katich presented Connolly with his baggy green on a bright Thursday morning before Australia lost the toss.
Connolly is Australia's fourth debutant in as many Tests after Sam Konstas, Beau Webster and Josh Inglis.
A member of the BBL team of the tournament, Connolly impressed selectors while bowling in the nets and at their preparation camp in Dubai ahead of the two-match Sri Lanka series.
He will also help Australia play recognised batters down to No.8 - Connolly averages 61.80 in first-class cricket and memorably made 90 on debut in the Sheffield Shield final last summer.
Sri Lankan Pathum Nissanka returns from a groin injury to replace Oshada Fernando as opening partner to Dimuth Karunaratne, who is playing his 100th and final Test.
The hosts have dropped legspinner Jeffrey Vandersay, one of few stand-outs from the first Test, and brought in a third off-spinner in Ramesh Mendis.
Lahiru Kumara replaces Asitha Fernando as the sole front-line paceman.
Victory in Galle will seal Australia's first series win in Sri Lanka since 2011 after they thrashed the hosts by an innings and 242 runs in the first match.
AUSTRALIA: Usman Khawaja, Travis Head, Marnus Labuschagne, Steve Smith (c), Josh Inglis, Alex Carey, Beau Webster, Cooper Connolly, Mitch Starc, Nathan Lyon, Matt Kuhnemann
SRI LANKA: Pathum Nissanka, Dimuth Karunaratne, Dinesh Chandimal, Angelo Mathews, Kamindu Mendis, Dhananjaya De Silva (c), Kusal Mendis, Ramesh Mendis, Prabath Jayasuriya, Nishan Peiris, Lahiru Kumara.
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