If it were me with a small area like that I would pick up a roll of sod at the local box box, and fix it with that.

Can I patch this up using the existing grass on my lawn (that is, not getting new sod)? And if so how would one do this?

When we make a new flower bed, put in new stepping stones, etc. I cut around the area where the sod is to be removed with a shovel. I then cut it into small section between 14 and 20 inches square.

Next using the shovel I carefully lift each section by lifting and pushing the shovel further under the section of sod being remove. At some point the section is becomes loose and can be moved to the new area.

Some times if I have a lot to remove, I use a side walk edger. I have had it for a long time. The blade is about 14 inches long and the cutting edge is serrated. You could also use a normal side walk edger to make the cuts. Even with this tool, I still use the shovel to lift the sod out.
I also use this method to re contour small areas of the yard. To create the area where I want the new drainage area, lift all of the sod, contour the area and then replace the sod. When done no one knows I have been there, but the water flow is going where I want.

This post was edited by knuttle on Sat, Jan 17, 15 at 9:09

Can I patch this up using the existing grass on my lawn (that is, not getting new sod)? And if so how would one do this?

Then what are you going to do about THAT bare spot?