Robert Henry Southwell, styled The Honourable from 1766, was an Irish politician.
He was the second son of Thomas Southwell, 1st Viscount Southwell and his wife Margaret Hamilton, daughter of Arthur Cecil Hamilton.
Southwell served as lieutenant-colonel of the 8th Dragoons. In 1776, he entered the Irish House of Commons for Downpatrick, representing the constituency until 1783. Southwell died at Clontarf, Dublin.