Nevrev was born to a family of merchants. His first art lessons came from his stepfather, who was a drawing teacher. At the age of 21, Nevrev entered the Moscow School of Painting, Sculpture and Architecture (MSPSA), where he studied under the Russian-Italian painter, Mikhail Scotti. Upon graduating in 1855, he was named a "Free Artist". In the 60s, Nevrev painted one of his masterpieces, "The Bargain" (1866), in which he depicted the sale of a serf. Some other paintings focused on criticisms of the Church.
He temporarily stopped working in the 1870s, for unknown reasons, but began painting historical art in the 80s. In 1881, he became a member of the Association of Travelling Art Exhibitions. His best work during this time was arguably his genre paintings, each exhibiting a human moral. From 1887 to 1890, he taught at the MSPSA and, in 1889, became curator at the Tretyakov Gallery.
In 1898, following the death of Pavel Tretyakov, he was offered the position of Director at the gallery, but declined, citing old age and poor health. At the age of 74, in great financial distress, he committed suicide by shooting himself at his estate near Mogilev.