Literally dealt with the exact same thing on our back patio after moving some playground stuff & sod. Pondered & researched french drains and all kinds of exotic methods.
Ended up just cutting the sod up about a foot out from our patio. Then once it was out, dug out an inch or two of dirt from underneath, put the sod back down so the dirt is slightly lower than the patio. Has worked almost perfectly, took like an hour.