The Dons had slipped to a 2-1 defeat in last week's first leg and they were unable to turn that around after a 1-1 draw at Pittodrie this evening.
The Kazakhstan outfit worked another chance moments later when Stanislav Lunin was slipped through on goal by Gohou, before seeing his effort blocked by Ward.
Aberdeen then thought that they should have been given a penalty when Peter Pawlett went down inside the box, but the referee waved away their appeals.
Hayes was soon involved again when he latched on to a flick from Pawlett and drilled a strike from the edge of the box inches wide of the post.
The visitors almost extended their lead in the tie just before half time when Mikhail Bakayev fired in a shot towards the corner that Ward saved at full stretch.
Following a goalless first half, Aberdeen started strongly after the break, with Niall McGinn drilling an effort into the side-netting.
However, it was Kairat who broke the deadlock in the 58th minute when Bakayev's effort was saved by Ward and Gohou turned in the rebound to give his side a 3-1 aggregate lead.
Aberdeen now needed two goals to take the game to extra time, and Shay Logan tried his luck with an attempt that was kept out by Plotnikov at his near post.
The hosts almost found themselves two goals down when Serginho crashed a strike against the bar, but McLean gave them some hope of salvaging a comeback when he forced the ball in from close range.
The goal provided a major boost for the Dons, and McGinn almost added a quick second when he burst into the box and drilled a low strike into the side-netting.
In the final minute of stoppage time Aberdeen came agonisingly close to taking the game to an extra 30 minutes as Paul Quinn's header was tipped over by the keeper, but it proved to be the last action of the match, as their European campaign came to a premature end.