Finch suffered the injury while batting during the team's 27-run loss at the Melbourne Cricket Ground, which saw India take an unassailable 2-0 lead in the three-match series.
Left-handed batsman Usman Khawaja has replaced Finch in the squad for Sunday's match and has also been included in Australia's one-day international squad for the tour of New Zealand.
"It is an honour to be asked to lead the side on Sunday, although the circumstances are far from ideal," Watson said in a Cricket Australia statement.
"We may have lost the series but there is a great deal to play for with the ICC World Twenty20 coming up and you can be sure we will be determined to do well in Sydney."
The 34-year-old Watson has previously captained Australia in a lone test and nine ODIs and is the country's second most experienced player in T20 internationals behind opening batsman David Warner.
(Writing by Sudipto Ganguly in Mumbai; Editing by John O'Brien)
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