Digging lawn to reach water line

So it seems the water line from city to my house is leaking under my lawn (so my responsibility) because when water was shut off the water meter still moved, so don’t think it would be a sprinkler line.

I’m going to dig the hole to the line to save money from paying the plumber to do it since home warranty coverage excludes digging but covers the pipe repair.

Is there any particular best practice to dig into the lawn so as to preserve the grass? Or do I just need to dig to find the pipe and try to just leave as much grass untouched as possible and then re-seed the plot that I had to dig up once it’s done?