Tebenikhin began his piano studies at the age of six with his father Vladimir Tebenikhin, himself a student of Lev Oborin. He continued his education at the Baiseitova Music College in Almaty with Janiya Aubakirova and Mikhail Balabitchev. From 1996 to 2003, he studied with Mikhail Voskressensky at the Moscow Conservatory. From 2004 to 2012, he studied at the Hanover Conservatory with Karl-Heinz Kämmerling. He won the Vianna da Motta Competition in Portugal in 1999.
In 1999 he won the Vianna da Motta Competition in Portugal, and in 2001 the Piano Competition in Le Mee sur Seine in France. In 2004 he was a prize winner at the Queen Elisabeth Music Competition in Brussels. In 2004 he won prize at the Glasgow International Piano Competition and at the 1st International Piano Competition in Panama City. "In 2006, "Amir earned 1st place at the ZF Music Prize in Friedrichshafen." In 2007, he won first prize at the 3rd Anton Rubinstein Piano Competition in Dresden. In 2011, he won the International German Piano Award in Frankfurt.
Tebenikhin has performed in Europe, Asia, Latin America, and the U.S. He has appeared at festivals in France, Germany, Belgium, and Russia. Tebenikhin has made radio and CD recordings (on the Naxos, Classical Records, Genuin, Cypres Label, K&K Verlagsanstalt labels, among others).