The pair, who finished ninth in Beijing despite being one of the favourites for gold, ended the medal race in fifth as Denmark’s Allan Norregaard and Peter Lang snatched the bronze.
The 2007 world champions were visibly disappointed to miss out on a medal but conceded that their chances of the podium were lost earlier in the week, rather than in today's medal race.
“It was always going to be tricky,” said Rhodes. “Ultimately I thought the Austrian guys had sailed a pretty epic race but unfortunately for them the Danish got a got a good shift up the left and got back to bronze.
“For us we came out with the aim of sailing a good race today and that was all we could really do. We felt like we did a reasonable job.
“We are reasonably pleased with today and the way we sailed, but ultimately this event was lost a bit longer ago. We were in the hunt this morning but we had the potential to put it out of doubt."
Gold and silver had already been decided with Australia’s Nathan Outteridge and Iain Jensen taking the title with New Zealand’s Peter Burling and Blair Tuke claiming silver.
Morrison and Rhodes were six points behind the Danish crew, who were third after the 15 races, and the bronze fight was a six-way battle with the Austrians narrowly missing out after winning the medal race.
“That’s just the nature of sailing really,” said Morrison. “We’ve done eight races here under the Nothe which is obviously very shifty and gusty.
“I think we’ve not really done ourselves justice on this particular course. On the bigger courses we’ve really sailed a good week.
“Obviously it is pretty gutting but what can you do? It’s done now and we’ve got to move on. It’s amazing the support we’ve got – our family are there which is really cool but just pretty gutting.
“Like Ben (Rhodes) said we threw it away, in the middle of the week – we should have been far better off on the points than we were but we’re not and that’s just how it is.”