Antoine-Curier had an effort saved by Jamie Langfield before the Frenchman struck the opener.
Mickael Antoine-Curier gave Hamilton the lead from close range 13 minutes before the half-time interval.
However, Antoine-Curier's left-foot effort from the edge of the box trundled towards home keeper Enckelman.
Mickael Antoine-Curier headed a reply a minute after the break, but Prince Buaben poked home after 59 minutes.
But parity was short-lived as Wesolowski combined with Antoine-Curier before drilling a shot beyond Langfield and into the net.
They worked an opening when the towering Elliott knocked on Routledge's free-kick Anderson brilliantly blocked Antoine-Curier's shot on the turn.
Richard Foster's clearance was collected by James McArthur, whose mis-hit shot found its way to Antoine-Curier and the striker slotted past Langfield.
Mickael Antoine-Curier was Accies' lone striker, but he had almost no support in the opening stages as Hearts held on to possession.
Anderson produced another crucial block in the 75th minute to prevent Elliott from getting on the end of Antoine-Curier's cross on the break.
Playing staff also laid-off included Brian Kerr, Mickael Antoine-Curier, Eric Paton, Dominic Shimmin, Njazi Kuqi, Paul McHale, Scott Fox, Colin McMenamin and Charlie Grant.
Michael Duberry and Steven Anderson kept Accies strikers Mickael Antoine-Curier and Damian Casalinuovo at bay but the visitors did manage a shot on target in the 30th minute.
Accies' top scorer Mickael Antoine-Curier made his team's starting line-up, despite concerns that a knee injury might preclude his participation, and had he not strayed offside an equaliser might well have resulted.
Paixao continued to pose a threat to United in the second half and the Portuguese player crossed invitingly for Mickael Antoine-Curier but the big striker could not direct his header on target.
However, the home side were back in the tie within seconds of the restart when Graham broke down the right, his cross found Antoine-Curier at the back post and the powerful header flew past Pernis.
Antoine-Curier played a lovely low ball from the wing into the path of Dougie Imrie, but the diminutive forward's poor first-touch as he tore unmarked into the penalty box took it safely into the arms of goalkeeper Dusan Pernis.
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