Losing finalist in Madrid last weekend, Swiatek looked again in imperious form in the tournament she won last year before going on to blitz the French Open.
The 21-year-old is trying for her third straight Rome title and made light work of Tsurenko after a slightly bumpy start.
Swiatek lost the first two games but then rattled off 12 in a row in a powerful demonstration of why she is world number one and favourite for Roland Garros.
Swiatek now faces either Donna Vekic or Ljudmila Samsonova in the fourth round after extending her winning streak in the Italian capital to 13 matches.
Her bid for a Rome hat-trick will be helped by five of the world's top 10 women players being eliminated early, world number eight Maria Sakkari the latest victim after she was beaten 7-5, 6-3 by Marketa Vondrousova.
Sakkari joins world number two Aryna Sabalenka, third-ranked Jessica Pegula, world number four Caroline Garcia and seventh ranked Ons Jabeur in being dumped out of the tournament.
In the men's draw Medvedev came through in straight sets against Ruusuvuori to claim his first ever win at the Foro Italico courts.
The former US Open champion won a second round match which was postponed to Sunday due to the torrential rain which hit the Italian capital on Saturday.
In much balmier conditions the world number three was dominant on a clay court surface he is famous for not liking.
Few would back him to go deep in Rome but his performances on clay have improved, reaching the quarter-finals at the Monte Carlo Masters.
Medvedev will face Bernabe Zapata Miralles in the third round.
Stefanos Tsitsipas will take on home hope Lorenzo Sonego after finishing off Nuno Borges 6-3, 6-3 in a match interrupted by Saturday's rain with the world number five already well on his way to victory.
