England, who have won the last two Ashes campaigns, lead the five-match encounter 2-0 and as holders will keep the urn in the event of a drawn series.
The hosts were looking for the draw but were in trouble at 37 for three when play was suspended after lunch and steady rain continued to fall until the end.
"It's a bit of a strange day. Today we had a little bit of luck and it's nice to retain the Ashes," said England captain Alastair Cook at an indoor presentation.
 "The feeling in the dressing room is a very pleasant one. Now we want to go on and win it (the series)," Cook said.
Australia would have fancied their chances of taking the remaining seven wickets in the final two sessions had the rain not intervened.
Play was eventually called off at 1539 GMT to the cheers of a smattering of hardy fans still left inside the Manchester ground, who later greeted the celebrating England players on the team balcony.
The teams next face each other at Chester-le-Street in Durham from Friday for the fourth test which will be a slight anti-climax after the literal damp-squib finish at Old Trafford.
England will still want to win the Ashes outright for the third straight time while Australia will at least look to make it 2-2 and head home with some confidence for the next series between the two starting in November.
"It's unfortunate. I certainly don't want to take anything away from England. It's the chance you take going 2-0 down," Australia skipper Michael Clarke said of the weather.
England's Joe Root was unbeaten on 13 from 57 balls after the opener tried to leave every wide delivery he could. Ian Bell was four not out having been struck on the glove and seen the ball balloon over the slips just before the second rain break.
Bustling fast bowler Ryan Harris struck twice for Australia before lunch to give the tourists real hope they could drag themselves back into the series after a 14-run loss at Trent Bridge and a 347-run humbling at Lord's.
 Rain was in the air from the start and play began half an hour late, leading Australia to declare on 172 for seven in their second innings and set England 332 to win.


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