Players of West Ham and Domzale line up before the game
Slaven Bilic's side put in an abject performance in what was their first competitive game of the season and were lucky the deficit was not even worse.
Bilic was actually banned from the touchline following his dismissal at the same stage of the competition last year against FC Astra.
West Ham's Mark Noble scores their first goal with a penalty
The Hammers went 1-0 down on 10 minutes when the referee adjudged Havard Nordtveit had handled in the box and awarded a penalty which Crnic converted.
Mark Noble restored parity from the penalty spot but West Ham went on to concede in the second half as Crnic struck again.
NK Domzale's Gaber Dobrovoljc in action with West Ham's Andy Carroll and Adrian
They failed to threaten the Domzale goal at all and could have lost by more.
As it stands it is all to play for in the second leg - which is also the first ever game at the club's new 60,000 seater Olympic Stadium home.
West Ham's Andy Carroll reacts
Here's how fans of West Ham, and some of their rival clubs, reacted to the performance and result on Twitter.
@whufc_official Most probably threw the game on purpose who wants to be Europa league ffs...— Cal (@yatesy9230) July 28, 2016
@whufc_official poor Result... Well you guys need to focus on the home game... would love to watch you guys play in the Europa— Omekannaya (@Sr_castic) July 28, 2016
@whufc_official how do you go through that 90 minutes and only use 1 sub? Bad performance but bad decisions before/during cost us.— Aaron Palmer (@Aaron270391) July 28, 2016
@whufc_official lucky to only be 1 down, defensively abysmal tonight, Slav needs to take charge next week, did nothing to change the game— T.Greenaway-Clissold (@TGC1982) July 28, 2016
@whufc_official Absolutely disgusting display. Who wants to join this shower. Gold + Sullivan all talk . Bilic niev— jockney (@dmwalker70) July 28, 2016
@whufc_official why don't we buy a right back and leave Antonio to what he's good at going forward he will be slaughtered in PL— Jim Mc Dermott (@jmd122) July 28, 2016
@whufc_official uninspired, not a single player looked threatening— Joe Pavlik (@J_P_Tres) July 28, 2016