The eighth-ranked Rublev hit 11 aces and never faced a break point against his Slovakian opponent, who had won their only previous meeting on hard court four years ago.
After his recent tournament wins in Hamburg and St. Petersburg, Rublev joined Novak Djokovic as the only players with four titles in the shortened 2020 season.
Headlining the event in Austria, Djokovic is set to play Filip Krajinovic in an all-Serbian opening-round match on Tuesday.
Djokovic, who won the event in his most recent appearance here in 2007, would meet Borna Coric in the second round.
Two wins would guarantee Djokovic the year-end No. 1 spot for a sixth season, matching the record set by Pete Sampras in the 1990s.
Coric, who lost the St. Petersburg final to Rublev last week, defeated Taylor Fritz 6-4, 6-4 after the American had 12 aces but dropped his serve at 4-4 in both sets.
Anderson, who won the event in 2018, closed out a 6-7 (2), 6-4, 7-6 (6) win and next plays Pablo Carreno Busta, who led Gael Monfils 6-1, 2-0 when the seventh-seeded Frenchman quit the match with an injury.
Amid tightened health and safety measures, 1,000 spectators were allowed in the 9,000-capacity Stadthalle arena.