How can I effectively replace sections of 7/8" plaster with drywall?
Shim out the framing to get the drywall flush. I've often used 5/8" drywall with no shims, depending on the plaster thickness and patch configuration, but otherwise shim it. 

1/8" hardboard is cheap and easy to work with. Or rip some scrap lumber to size. Staple it in place and be happy. 

Update: After seeing your photo, I suggest re-cutting your edges with a circular saw and abrasive blade set to depth. You really want a nicer shape to work with. Snap or trace straight lines, level and down the center of studs. Use ear, eye, and respiratory protection.