The hosts were chasing 215 from 61 over, and were 53-2 after 11 when the weather brought the match to a premature end.
Australia captain Michael Clarke declared with his side 160-8 in, following Kemar Roach's second five-wicket haul for combined figures of 10-146.
The tourists won the first Test by three wickets, and resumed the day on 73-3. However, they lost overnight batsmen Ricky Ponting and Clarke to the Windies' Fidel Edwards and Darren Sammy respectively.
Roach followed this up by recording his second five-wicket haul of the day for combined figures of 10-146.
The Windies slipped to 13-2 after lunch but looked on course to their first test win against the Australians since in almost a decade before the elements intervened.
The result sees the tourists maintain their 1-0 lead in the three-match series ahead of the final Test on Monday (April 23).