This may be a bit of a surprise, but according to some news sources (CNN, BBC), Buttigieg came in first and Biden fourth with Sanders and Warren 2nd and 3rd, after more than half the votes were counted:
The Iowa Democratic Party said results from 62% of precincts show Mr Buttigieg was ahead on 26.9%, followed by Bernie Sanders on 25.1%.
Elizabeth Warren was third on 18.3% and Joe Biden fourth on 15.6%.
Amy Klobuchar was 5th with 12.6%. The rest of the candidates (led by Yang) were much farther back (with 1.1%).
The whole thing is rather complicated because the Iowa primary seems to resemble a bit the voting system for the Bundestag:
Because Iowa precinct holds not one but two rounds of preference expression, or alignments, caucusgoers' second choices are more important than ever before.
If a caucusgoer's first-choice candidate doesn't break 15% of the vote on the first alignment, they can either switch their preference to a candidate who is viable in their precinct, be an uncommitted caucusgoer, or try to combine forces with other caucusgoers to make their first-choice candidate viable.
A caucusgoer whose first-choice is viable after the first alignment cannot, however, change their preference, meaning candidates can only gain and not lose votes in the second alignment. [...]
A candidate must break 15% of the vote in a given district to win any delegates from that district at all.
There are also five pledged delegates statewide for party leaders and elected officials, and nine at-large delegates. Those are allocated based on the statewide popular vote, meaning its possible for candidates to win district-level delegates but not statewide delegates.
(The Iowa system showcased with a [rather lengly] chart over here.)
With regard to the 1st preference, after counting slightly more of the votes (65%), other media sources announced that Yang did substantially better (5.5%) and that Sanders overtook Buttigieg... but not after taking into account the 2nd preference and the translation of votes into delegates. The less detailed articles I found/mentioned first apparently only announced these delegate-equivalent results. Below is the "1st preference" in the Iowa primary (insofar), which is fairly different...
So how far off are these results from the average polling in Iowa in the months preceding the primary? (This is a bit open-ended, because I don't know if pollsters in Iowa tried to emulate/compensate for the complicated voting system over there, or not, when they did their polling. So, answers should try to clarify/specify that too.)