Just Google "How to unclog a toilet without a plunger" and you'll see this same tip/trick posted on a variety of sites including Youtube.
My own personal experience involved my oldest son jamming up the plumbing with what was likely a very liberal amount of moon wipe. Yes, that's toilet paper. I went at the clogged fixture with my trusty plunger believing it'd be only a matter of time. After-all, I've never failed before. After several attempts over an hour I did nothing but fail. So I began googling for ideas before running out for the next step up in toolage - toilet auger. I found several sites describing this method:
(1) Heat a large container of water (2-3L) but do not boil.
(2) While the water heats pour about a 1/4 cup of dish soap in the toilet. If you can plunge out most of the water before hand, do so before adding the soap.
(3) Pour the heated water into the bowl from waist height at a steady rate.
(4) Let sit for 10 minutes.
(5) Try and flush. At this point I was 100% successful. The clog disappeared instantly. Been flushing perfectly ever since.