From the e-visa Website:
I have a valid Supporting Document (Valid visa OR valid residence permit from one of the Schengen Countries, USA, UK or Ireland). E-visas are not accepted as supporting documents.
It's a self-check based application, meaning you have to cross the checkboxes to proceed. If you don't fulfil all of the criteria (in this case, because your US visa is expired), youe e-visa won't be valid, and if you get a strict officer (particularly likely at Istanbul-Atatürk airport), you will be refused entry and sent back, and the airline will be heavily fined for transporting you.
Turkish border police do not mess about, and a lot of them speak zero English at all, meaning they'll hardly understand what an I-797 is, nor will they bother to find out.
So, in essence, you need a regular visa, despite what the embassy says.