The result left the Croatians second on 17 points ahead of the last round of games, two behind leaders Iceland who moved above them with a 3-0 win in Turkey.
Third-placed Ukraine also have 17 points but an inferior goal difference after a 2-0 win at bottom team Kosovo. Their clash with Croatia in Kiev on Monday is likely to decide who clinches the runners-up spot.
Both teams can still win the group only in the unlikely event that Iceland, aiming for their second successive major tournament after reaching the Euro 2016 quarter-finals, slip up at home against Kosovo.
The outcome also piled more pressure on Croatia coach Ante Cacic, who was so shocked that appeared to forget who his team played at the post-game news conference.
"Iceland deserved the draw they snatched," he said, confusing the Finns with Iceland who are favourites to clinch an automatic berth for the tournament in Russia.
"Our final ball and the finishing keeps letting us down in this campaign but we now have to regroup.
"We are very disappointed because we've dropped two vital points and made things complicated for ourselves. The lads played their hearts out but lost their focus in the closing stages."
Late substitute Pyry Soiri celebrated his international debut for Finland with a fine finish which silenced the home crowd, who then jeered the Croatians off the pitch after a tepid performance.
The 23-year-old's volley from eight metres cancelled out Mario Mandzukic's 57th-minute opener after the home side missed a string of chances in the first half.
The best opportunity fell to centre back Domagoj Vida whose header was cleared off the line by Rasmus Schueller after left back Josip Pivaric and winger Ivan Perisic also came close.
Captain Luka Modric, making his 100th international appearance, fired inches over the bar from 20 metres in the second half before the Finns threatened as Juhani Ojala's header forced an acrobatic save from goalkeeper Danijel Subasic.
Mandzukic scrambled the ball in from point blank range soon afterwards, with Pivaric's cross diverted into his path by Finnish defender Niklas Moisander.
The Croatians sat back in the closing stages and were stunned at the death by Soiri who took advantage of static defending and blasted his volley past Subasic.
(Writing by Zoran Milosavljevic in Belgrade; Editing by Christian Radnedge)
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