Panyukov made his debut in the Russian Premier League on 21 July 2012 for FC Dynamo Moscow in a game vs FC Volga Nizhny Novgorod.
He was the top scorer of the Under-21 Russian Premier League competition of Spring 2012, scoring 13 goals in 13 games.
On 31 October 2012, he scored his first goal for the main Dynamo squad, giving them a 2–1 victory over FC Khimki in the 2012–13 Russian Cup Round of 16 game.
Panyukov joined FC Khimki on loan for the second part of 2012–13 season.
He scored his first league goal for Dynamo on 18 August 2013, giving them an equalizer with 7 minutes left to play in an away game vs FC Krasnodar (the game ended 1–1).
For the second part of 2013–14 season Panyukov went on loan to PFC Spartak Nalchik.
Panyukov joined Lithuanian A Lyga contenders Atlantas on loan before 2015 season. According to his agent, he chose Lithuanian club due to possibility to get a lot of playing time and be team leader.
He quickly managed to adapt in his new side and was lethal in front scoring 11 goals in the first round of the league becoming Player of the Round.
After his contract with Dynamo Moscow expired Andrei choose to stay in Lithuanian and signed one year deal with Atlantas. Later he extended contract with the club until the end of 2018.
For the 2015–16 season, Panyukov joined Ajaccio on loan, with an option to buy.
Panyukov was loaned to the Portuguese club S.C. Braga before the 2016–17 season, with an buy-out option. However, he only made 6 appearances for Braga B and none for the main squad, and on 30 January 2017, the club announced that his loan has been terminated.
Andrei Panyukov joins again his former club FK Atlantas for six months before to sign in Russian Premier League with Zenit FC. He scores 11 goals in 14 matches.
On 10 August 2017, he returned to Russia, signing a 3-year contract with FC Zenit Saint Petersburg.