Passport expiry

You should be fine. You also won't need a visa as a tourist visiting Schengen nations (Austria is Schengen) for under 90 days. According to the Austrian embassy website,

your passport must be valid 3 months beyond your planned date of
departure from the Schengen area

Thank you very much Continental. I tried calling the Austrian embassy in the US but I could not get a person on the phone.

This topic comes up regularly. The issue is not your return flight, it is not even what the letter of the law is, it is what your airline says on departure from the US when you leave for your trip (unless you are already in Austria, then no issue)

The airline has a carraige responsibility to assure that all passengers are qualified for entry to the destination. If you are turned away at the border due to a Passport issue, they have to haul you back at their expense, plus be subject to a hefty fine. As such, they use a database or system to verify your passport, or use a broad stroke of 6 months validity (for Europe) from the time of departure.

As such, I would contact your airline and talk to them. Lets face it, if they refuse you boarding, it does not matter what an Embassy says, what border control does when you arrive, or even what people on here say.

If you do a search of past posts, you will see more discussion, as well as some stories of people who showed up at the airport and did not get on the flight. If you have time, apply for a new one, you have an express option, as well as an option to go in person and get it within a couple days.

This link accesses the database most airlines use to determine passport validity:
https://tinyurl.com/y9vcubmg

Using the data in your posting, it says your current passport is kosher.

Paul has it correct, it is what the airline check-in person says on your departure:
3 months from your planned return date if have they been properly trained and are in a good mood.
6 months from your departure date if you are unlucky.

You will always be allowed back into your own country, even if the passport has only 1 day left.

Thanks for your responses. I checked the link posted for United travel requirements, it happens I am traveling on United/Lufthansa,
Summary
Yes, The travel documents held by the traveler are sufficient!
I checked my passport, it is 5 months 12 days.
My flight is tomorrow, I don't know what else I can do at this point before checking in.
Thanks everyone for your help.