Combine corned beef and spice packet contents, water, onion, carrots, celery, and salt together in a large pot or Dutch oven; bring to a simmer, skimming off any foam that rises to the surface.
Cover the pot, reduce heat to low, and simmer until meat is almost fork-tender, about 3 hours. Add potatoes and simmer, uncovered, until potatoes are almost tender, about 30 minutes more.
Place cabbage pieces on top of and around meat, cover the pot, and simmer until cabbage is tender, 20 to 30 minutes more.
Remove meat to a cutting board and let rest 10 to 15 minutes. Cut across the grain and serve in a bowl; ladle vegetables and broth over the top.
