This works for me. It closes with a backslash, i.e. Your closing tag.
For example, you have <div>my div< and after entering / you get <div>my div</div> .
To enable, add the following object to the array in the key binding file (Sublime Text> Preferences> Key Bindings).
{ "keys": ["/"], "command": "close_tag", "args": { "insert_slash": true }, "context": [ { "key": "selector", "operator": "equal", "operand": "(text.html, text.xml, meta.jsx.js) - string - comment", "match_all": true }, { "key": "preceding_text", "operator": "regex_match", "operand": ".*<$", "match_all": true }, { "key": "setting.auto_close_tags" } ] }
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