Novikov was born in Gorky, Russian SFSR (now Nizhny Novgorod, Russia). He grew up in a family of talented mathematicians. His father was Pyotr Novikov, who gave the negative solution of the word problem for groups. His mother Ludmila and maternal uncle Mstislav were also important mathematicians.

In 1955 Novikov entered Moscow State University (graduating 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 the PhD). In 1965 he defended a dissertation for the Doctor of Science in Physics and Mathematics degree there. In 1966 he became a Corresponding member of the USSR Academy of Sciences.

Novikov's early work was in cobordism theory, in relative isolation. Among other advances he showed how the Adams spectral sequence, a powerful tool for proceeding from homology theory to the calculation of homotopy groups, could be adapted to the new (at that time) cohomology theory typified by cobordism and K-theory. This required the development of the idea of cohomology operations in the general setting, since the basis of the spectral sequence is the initial data of Ext functors taken with respect to a ring of such operations, generalising the Steenrod algebra. The resulting Adams–Novikov spectral sequence is now a basic tool in stable homotopy theory.

Novikov also carried out important research in geometric topology, being one of the pioneers with William Browder, Dennis Sullivan and Terry Wall of the surgery theory method for classifying high-dimensional manifolds. He proved the topological invariance of the rational Pontryagin classes, and posed the Novikov conjecture. This work was recognised by the award in 1970 of the Fields Medal. From about 1971 he moved to work in the field of isospectral flows, with connections to the theory of theta functions. Novikov's conjecture about the Riemann-Schottky problem (characterizing principally polarized abelian varieties that are the Jacobian of some algebraic curve) stated, essentially, that this was the case if and only if the corresponding theta function provided a solution to the Kadomtsev-Petviashvili equation of soliton theory. This was proved by T. Shiota in 1986, following earlier work by E. Arbarello and C. de Concini, and by M. Mulase.

Since 1971 Novikov has worked at the Landau Institute for Theoretical Physics of the USSR Academy of Sciences. In 1981 he was elected a Full Member of the USSR Academy of Sciences (Russian Academy of Sciences since 1991). In 1982 Novikov was also appointed the Head of the Chair in Higher Geometry and Topology at the Moscow State University.

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 is also a Distinguished University Professor at 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 is one of just eleven mathematicians to have received both the Fields Medal and the Wolf Prize.