Ondrej Duda put Slovakia ahead in the first half but Romania's Razvan Marin equalised with a penalty before the interval.
With all four teams in the group earning four points from their three games, Romania finished top having scored more goals than second-placed Belgium.
Slovakia went through in third place thanks to a superior goal difference to fourth-placed Ukraine.
It was a memorable moment for Romania, who are into the knockout stages of the Euros for the first time in 24 years.
Romania had only qualified once for the last 16 in their five Euro appearances, but Edward Iordanescu's tenacious side have ended that long drought
"It's incredible. I want to congratulate my players, we found a spirit. We came back once again, we never give up," Iordanescu said.
"These are things you experience once in a lifetime. We have written history."
Slovakia are into the knockout stages of a major tournament for the third time.
They reached the last 16 in the 2010 World Cup and made it to the same stage at Euro 2016.
"It was a great performance," Slovakia boss Francesco Calzona said. "It means a lot. We were 55th in the FIFA rankings when I started, so to get here makes me very proud.
"We have world-class opponents ahead but it will be a celebration for us."
Knowing a draw would send both teams through, cynics had suggested they would play out a mutually beneficial stalemate.
But there was no sign of Romania settling for a point as they made a fast start.
They threatened an early goal when Andrei Ratiu's stinging strike was pushed away by Martin Dubravka, with the rebound lashed wildly over by Ianis Hagi.
Slovakia were inches away from snatching the lead as Lukas Haraslin's free-kick went through a crowd of players and almost crept in at the far post.
It was a warning Romania failed to heed and Slovakia went in front in the 24th minute.
Juraj Kucka whipped in a deep cross that caught the Romania defence standing statuesque, leaving the unmarked Duda free to power his header past Florin Nita from eight yards.
Romania's response was emphatic as they drew level in the 37th minute thanks to an incisive raid from the son of former Barcelona and Real Madrid star Gheorghe Hagi.
David Hancko's foul on Hagi was adjudged a penalty by VAR after referee Daniel Siebert initially ruled the offence had taken place just outside the area.
Marin stepped up to drill the spot-kick high into the roof of the net to the deafening delight of the hordes of Romania fans who turned the arena into a frenzied mosh-pit.
The searing Frankfurt heat gave way to a heavy storm in the second half as thunder, lightning and rain swirled around the stadium.
It was a suitably dramatic backdrop to a tense clash that was close to swinging Romania's way when Marin forced a good save from Dubravka before Denis Dragus skimmed the roof of the net.
Haraslin's curler flashed past the far post as the storm abated, but, while there was no winner, both sides ended up celebrating their progress to the knockout stage.
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