Jelistratow made his debut in Salt Lake City in October 2008 in the Short Track World Cup and finished in 19th place over 1000 m and 1500 m. At his first Olympic participation in Vancouver in 2010 , he came 29th over 1000m and 24th over 500m and 1500m. In March 2010 he achieved 31st place in the all-round competition and over 1000 m and 30th place over 500 m and 1500 m at the 2010 World Championships in Sofia . In the 2010/11 season he won his first World Cup victory in Changchun over 500 m. At the 2011 European Championships in Heerenveen , he won silver with the relay. In the following season he reached two second places in the World Cup with the relay in Saguenay and Nagoya and a third place over 1500 m in Dordrecht . At the European Championships in Mladá Boleslav in 2012 , he won the silver medal with the relay and the bronze medal in the all-around with second places over 1500 m and 3000 m. In the 2012/13 season he came in twice in third place and twice in second place in the World Cup. In Sochi he won his second World Cup victory in the relay. At the 2013 European Championships in Malmö , he won the gold medal in the 5000 m relay. In addition, he was fifth in the all-around competition and took third place in the 1000 m and 3000 m respectively. In March 2013 he won silver with the relay at the World Championships in Debrecen .
In the 2013/14 season, Jelistratow came second in the World Cup with the relay twice. In January 2014 he won the silver medal in the all- around competition at the 2014 European Championships in Dresden and the gold medal with the relay. He achieved second place over 1000 m and first place over 1500 m in the individual races. At the 2014 Winter Olympics in Sochi , he was Olympic champion with the relay over 5000 m. At the beginning of the 2014/15 season, he finished second in the 1000 m and first place in the relay in Salt Lake City . In the further course of the season he won another three races and took second place once. At the European Championships in Dordrecht in 2015 , he won the bronze medal in the all-around competition and the gold medal with the relay. He was third over 500 m, 1000 m, 3000 m and second over 1500 m. In March 2015 he became world champion over 1500 m in Moscow . He finished the season in first place in the 1000 m World Cup. In the following season he finished three times third and three times second in the World Cup singles. He also won in Nagoya over 1000 m and in Dresden and Dordrecht each with the relay. As in the previous year, he won the World Cup over 1000 m and also achieved sixth place in the World Cup over 1500 m. At the European Championships in Sochi in 2016 , he won the gold medal in the all-around competition. He was second over 500 m and first over 1000 m and 1500 m. After third place each over 1500 m in Calgary and Salt Lake City and second place over 500 m in Salt Lake City, he came in second and once in third place in the World Cup singles. In Dresden he won the relay. At the European Championships in Turin in 2017 , he won the silver medal with the relay and, as in the previous year, the gold medal in the all-around competition. In the individual races he took third place over 1000 m and first place over 1500 m and 3000 m. His best results at the 2017 World Championships in Rotterdam were fifth place over 1000 m and fourth place with the relay. He finished the season in third place in the 1500 m World Cup.
In the 2017/18 season, Jelistratow won the bronze medal in the all- around event at the 2018 European Championships in Dresden and the silver medal over 1000 m, 1500 m and with the relay and at the 2018 World Championships in Montreal the bronze medal over 500 m, 1500 m and over 3000 m. At the highlight of the season, the 2018 Winter Olympics in Pyeongchang , he won the bronze medal in the 1500 m. He also ran there on the 20th place over 500 m and on the sixth place over 1000 m. In the following season he won with the relay in Turin and with the mixed relay in Dresden and Turin. In addition, he achieved four top ten placements over 1500 m and tenth place in the World Cup over 1500 m. At the European Championships in Dordrecht in 2019 , he won the bronze medal in the all-around, over 1500 m and with the relay, and the gold medal over 1000 m. In March 2019 at the World Championships in Sofia he won the bronze medal in the all- around event and over 1000 m and the silver medal over 3000 m. After finishing one m above 1500 and with the season in Salt Lake City earlier in the season 2019/20, he won in Montreal over 1000 meters and finished there and in Nagoya each third place in the relay and in Nagoya third place 1000 m. This was followed by first place with the relay and second with the mixed relay in Shanghai and second place in Dresden over 1500 m, with the relay and the mixed relay. At the European Championships 2020 in Debrecen he won the bronze medal in the 500 m, 1000 m and all-around competitions and the gold medal with the relay. He finished the season in fifth place in the 1500 m World Cup and third in the 1000 m World Cup.
In March 2016, he was proven to have taken the banned substance meldonium .