Alyoshin, responsible for elite and recreational sports, became the second high-ranking official to quit after Russia's dismal showing at last month's Winter Olympics in Vancouver.
Leonid Tyagachyov resigned as president of the Russian Olympic Committee (ROC) on March 3, two days after President Dmitry Medvedev ordered top sports officials to quit or be fired.
Russia, a traditional Winter Olympics superpower, produced its worst display, winning 15 medals but only three golds.
Sports Minister Vitaly Mutko said he would stay in the job, vowing to improve fortunes in 2014 when the country hosts the next edition in Sochi.
The ROC's executive board will meet on Wednesday to discuss Tyagachyov's successor.