After Australia were all out for 203 in reply to Sri Lanka's first innings total of 117, the hosts then lost opener Kusal Perera for just four when he was lbw to a ball from paceman Mitchell Starc.
After Perera's dismissal, a sudden downpour forced the players to take an early tea break on the second day.
But the deluge soon made it impossible for the match to resume and the umpires called stumps early for a second day in a row.
Earlier, Australia had gone on to build up a lead of 86 runs over their hosts, which could prove crucial in a low-scoring match.
Veteran Sri Lankan spinner Rangana Herath and the debutant Lakshan Sandakan took four wickets each while seamer Nuwan Pradeep picked up two wickets.
Adam Voges top-scored for the Australians with 47, meaning no player from either side managed to score a half-century in the first innings.