We're on vacation in Cancun and the hotel room was missing a corkscrew. I called room service and while waiting I found this video. It really works. Be careful, after the cork is sticking out a bit, try to pull it out.
I made the mistake of going all the way until the cork popped out. I had to then search for "removing red wine from fabric" and learned that club soda really works.
Housekeeping never showed up with that corkscrew...