Krishna was born in the Coastal Andhra region of Andhra Pradesh, India. He received a BTech degree in electrical engineering from Indian Institute of Technology, Kanpur in 1985. He subsequently moved to the United States to earn a PhD in electrical engineering from the University of Illinois at Urbana–Champaign in 1991. Krishna joined IBM in 1990, at IBM's Thomas J. Watson Research Center. He worked with the company in numerous positions, and was a driving force behind IBM's $34 billion acquisition of Red Hat. He was appointed IBM's CEO in January 2020. In 2021, he was named by CRN as the year's "Most Influential Executive".