Despite being a Spartak Moscow fan, Alenichev debuted 1991 for Moscow rivals Lokomotiv, where he played four years before moving to Spartak, where in five years he won three Russian leagues and two cups, and was also elected Russian player of the year in 1997. He won a transfer to Serie A side AS Roma, played 21 matches in the first season, but after only seven matches played, he moved to AC Perugia in December 1999. His stint in Italy overall proved to be unsuccessful and he was eventually considered to be one of Italian football's biggest foreign flops.
In 2000 he was bought by FC Porto, where he made a good first impression (scored the tying against Sporting in the Portuguese Super Cup), and made a good first season where Porto captured the Portuguese cup 2–0 against CS Marítimo, Alenichev scoring the second goal. In the following season, Alenichev suffered some animosity from new Porto coach Octávio Machado (the same happened with compatriot Sergei Ovchinnikov) and spent most of the first half of the season sidelined, under the shadow of Deco. As soon as Octávio was fired and replaced with José Mourinho, Alenichev's luck changed. Although he still didn't play in the starting eleven, he was usually the first player to jump from the bench, especially when Mourinho passed from a 4–3–3 to a 4–4–2 formation. A starting player in the UEFA Cup final and mid-game substitute in the Champions League final, Alenichev scored on both: against Celtic he scored the second goal, following a pass from Deco, and against AS Monaco he closed the scoreline with a powerful volley shot following a deflected through cross from Derlei. This made him one of only three players to score goals in two consecutive Cup Finals of different European competitions, the others being Ronald Koeman and Ronaldo.
During Euro 2004, where he played all three matches Russia played, he announced his desire to return to Spartak Moscow, and in appreciation for the services done for the club, the FC Porto board made no objections to the transfer.
On 8 April 2006 Sport-Express published Alenichev's interview containing severe criticism of Aleksandrs Starkovs, Spartak's head coach at the time. Following that, Alenichev was fined, dismissed from the first team, transfer listed 14 April and on 10 September his contract was finally terminated by mutual agreement. This became the end of Alenichev's football career.