In 2002 he received the State Prize of the Russian Federation in the field of science and technology. In 2011 he was awarded the Prize of the Government of the Russian Federation in the field of science and technology. He also received the Orders of the Red Banner of Labour (1986), Honour (1999) and Friendship (2009).
The Royal Society of Chemistry awarded Moiseev the Centenary Prize for 2006/7. In 2012, he was awarded the Demidov Prize for his contribution to the chemistry of organoelement compounds, petrochemistry, and carbene chemistry, and the RAS Chugaev Prize for his work on coordination compounds in industrially important redox reactions. In 2013 he received the RAS Mendeleev Medal for outstanding work in the field of catalysis and energy-saving technologies.
He became a corresponding member of the Academy of Sciences of the Soviet Union in 1990, and an academician of the Russian Academy of Sciences in 1992. He was a full member of the Academy of Sciences, Arts and Literature in Paris, the European Academy of Sciences and Arts, and the Academia Europaea.