Group winners Germany, who have scored nine without reply in the group stage, had secured a quarter-final spot with a game to spare by beating Spain 2-0 and will face Austria on Thursday.
Germany imposed their dominance over Finland from the start, with striker Alexandra Popp (31) having their first attempt inside a minute, but Anna Signeul's Finland side defended well and had some luck to avoid their opponents taking an early lead.
It took 40 minutes and 20 shots at goal for the record eight-time champions to score against lower-ranked Finland, who did not qualify for the previous tournament and have now been winless in their last eight Euro finals games.
Finland goalkeeper Katriina Talaslahti (21) was not without merit either. Stepping in for Tottenham Hotspur keeper Tinja-Riikka Korpela (36), who was ruled out of the match with COVID-19, the 21-year-old tried everything to stop Finland from conceding.
Germany's well-deserved breakthrough, in a repeat of the Euro 2005 semi-final, arrived five minutes before half time when defender Sophia Kleinherne (22) headed home a precise cross from the right by midfielder Giulia Gwinn (23).
The Group B winners doubled the lead three minutes into the second half with another header as captain Popp became the first German - and second player ever - to score in all three Euro group games.
England's Beth Mead was the first to reach that mark on Friday, in their record 8-0 win against Norway.
Substitute Nicole Anyomi (22) then lashed the ball home in the 63rd minute from the edge of the box with a wonderful first touch for her first international goal at senior level.
"It was a hard piece of work," Popp told reporters.
"We were clearly the better team, but you could tell that our strength had waned a bit for the time being. But the team can be forgiven for that."
"First and foremost, the win counts. This is a starting position that we've earned. There are minor things to criticize, but we'll fix them before the quarter-finals," Kleinherne added.
Group B runners-up Spain, who beat Denmark 1-0 with a last-minute goal, will meet Group A winners England in Brighton in the first quarter-final on Wednesday.
Ranked 29th in the world, Finland's hopes of reaching the quarter-finals were ended by a 1-0 loss to Denmark on Tuesday after their opening defeat by Spain.
"We did really well against a very strong German team," Finland coach Anna Signeul said.
"We had (new) players on the pitch today. The players showed today that we have a strong squad... They made themselves and Finland proud - although we are, of course, not happy with the results."
