The Scottish side had been heading for a shock defeat in the first leg of their Europa League first qualifying round tie at Rhyl's Belle Vue stadium after a Greg Taylor own goal in the 75th minute.
But Killie, backed by the majority in the 3,000-capacity stadium, rallied and levelled with an Eamonn Brophy penalty eight minutes from time.
The visitors then continued to press and grabbed victory in injury time, when Findlay headed home from a corner.
It was harsh on Connah's Quay, who had battled stoically throughout in the mould of their manager Andy Morrison, the former Manchester City defender, but the visitors were by far the dominant side.
Alessio had named a strong team and they threatened early on, with Brophy having an effort blocked and Rory McKenzie firing over, but Connah's Quay prevented them getting into their stride.
Nomads' Callum Morris escaped a booking for a heavy challenge on Alan Power, but Callum Roberts did go into the book for a late tackle on Taylor.
Kilmarnock had a good spell after half an hour as Stephen O'Donnell had a shot tipped wide and then drove another effort off target following a blocked McKenzie shot. Brophy also showed good control in the area before shooting across goal.
Killie upped the tempo after the break and a Chris Burke cross flashed through the Nomads area.
Brophy looked certain to score when he got on the end of another Burke cross, but Nomads keeper Lewis Brass – who had exchanged words with some Killie fans after the half-time whistle – brilliantly clawed away from under the bar.
Alessio showed signs of frustration at his side's failure to take chances as he threw a water bottle into the ground.
Kilmarnock opened up the hosts again with some neat play on the right, but O'Donnell's cross found nobody. Alessio sent on new signing Mohamed El Makrini in and attempt to break through, but it was Connah's Quay who snatched the lead.
Roberts broke clear down the left and his cross was turned into his own net by Taylor.
That prompted a strong response and Brophy struck from the spot after Roberts bundled Burke over in the box.
There was still time for a winner, and Findlay delivered when he headed into the far corner in the 91st minute.