Novikov was born in Gorky, Russian SFSR (now Nizhny Novgorod, Russia). His father was Pyotr Sergeyevich Novikov, a talented mathematician, who gave the negative solution of the word problem for groups. His mother Ludmila and uncle Mstislav were also important mathematicians.
In 1955 Novikov entered Moscow State University (graduated from it in 1960). Four years later he received the Moscow Mathematical Society Award for young mathematicians. In the same year he defended a dissertation for the Candidate of Science in Physics and Mathematics degree at the Moscow State University (it is equivalent to PhD).
In 1965 he defended a dissertation for the Doctor of Science in Physics and Mathematics degree there. member of the USSR Academy of Sciences, Member of the Division of Mathematics, including applied mathematic.
In 1966 he became corresponding Member of the Division of Mathematics since July 01, 1966. Since 1971 Novikov has worked at the Landau Institute for Theoretical Physics of the USSR Academy of Sciences.
He finally was elected Academician of the Division of Mathematics since December 29, 1981 (Russian Academy of Sciences since 1991).
In 1984 he was elected as a member of Serbian Academy of Sciences and Arts.
As of 2004, Novikov is the Head of the Department of geometry and topology at the Steklov Mathematical Institute. He also teaches at the University of Maryland, College Park and is a Principal Researcher of the Landau Institute for Theoretical Physics in Moscow.
In 2005 Novikov was awarded the Wolf Prize for his contributions to algebraic topology, differential topology and to mathematical physics. He became one of just eight mathematicians who received both the Fields Medal and the Wolf Prize.