Svatkovskaya's first World Cup competition was the 2013 Irina Deleanu Cup in Romania. She won the bronze medal in All-around behind Ukrainian Alina Maksymenko, gold in Clubs and Ribbon, and bronze in the Ball final. She was the all-around bronze medalist at the 2013 Pesaro World Cup. At the 2013 Minsk World Cup, she won silver in All-around ahead of Belarusian Melitina Staniouta, gold in hoop, silver in ball, and bronze in the ribbon final. Svatkovskaya's first senior Europeans were the 2013 European Championships in Vienna, Austria. She won gold in her sole apparatus, hoop, and together with her teammates (Yana Kudryavtseva and Margarita Mamun) won the team gold medal for Russia. An ankle injury kept her out of the 2013 World Cup series held in Saint Petersburg, Russia. Svatkovskaya returned to competition at the 2013 Grand Prix Brno and won the silver in all-around ahead of Melitina Staniouta. In the event finals, she won gold in hoop and clubs, silver in ball and placed eighth in ribbon. At the 2013 Grand Prix Final in Berlin, she placed fourth in the all-around behind Miteva and qualified to two event finals, winning gold in clubs and finishing sixth in ribbon.