A 49-year-old woman suffered an injury to her shoulder, while a 72-year-old man was treated for a gash on his head, officials confirmed.
Reports said police went on to drive the bear out of town and into a forest.
Videos posted on social media showed the bear bounding along a road, and in one, lunging at a man on the pavement.
The news comes a day after a 31-year-old Belarussian woman apparently fell to her death trying to escape a brown bear in the nearby Low Tatra mountains.
She had been walking with a male companion through an area of thick forest and steep ravines when they were set upon by the bear.
According to the man, he and the woman fled in different directions. Her body was later discovered by a search dog shortly after he called for help.
There have been a number of bear attacks in Slovakia in recent years including one fatal attack in 2021, reported at the time to be the first in Slovakia for a century.
Better environmental protection in Central and Eastern Europe since 1989 has meant bears have returned to their natural habitats across the Carpathian mountain range, which stretches up from Romania through western Ukraine and on to Slovakia and Poland.
Some members of Slovakia's populist-nationalist coalition have called for looser protection for bears so the rising numbers can be regulated by hunting.
The Environment Ministry said that together with Romania it would present a proposal at the next EU Council of environment ministers to reclassify bears on the protected species list, as their numbers mean they are no longer endangered and can be selectively culled.
Researchers however say there has been no explosion in Slovakia's estimated bear population, which they say remains stable at around 1,275.
It's a shame those people had to go through that. I'm glad they are alive.