Fulham almost saw their hard work undone with the last kick of the game when substitute Zhilyaev blasted over the bar from eight yards.
In the 10th minute, Aberdeen's Michael Paton should have done better with a shot that he blasted over the bar from 12 yards out.
BBC: Hamilton's David Elebert (left) tussles with Sone Aluko
Having done well to control a clipped pass from Steve Davis, Miller blasted over the bar in three minutes when DaMarcus Beasley was in a more advantageous position.
Scott Arfield could have put the match beyond the hosts in injury time, but blasted the ball over the bar.
Following their first corner in the sixth minute the ball broke to the edge of the box to Kilmarnock midfielder Mark Reilly, who steadied himself but blasted the ball harmlessly over the bar.
Substitute Roque Santa Cruz blasted an effort over the bar after finding space on the edge of the box, but Howard was seldom called into making a save.
After a scramble in the visitors' box, Johnson turned and blasted a shot narrowly over the bar.
Danny Grainger's 20-yard free-kick forced United keeper Dusan Pernis into a save in the build-up to half-time, then Morais blasted a decent ball over the bar following a goalmouth scramble.
Donnelly passed up a chance to increase his side's advantage when he blasted a late penalty over the bar.
John Devine saved a Vincent Corey goal chance while Corey also blasted a great last-ditch goal effort over the bar.
Imrie blasted over Esson's bar five minutes into the second half, after an impressive burst of pace by Hasselbaink had first set up Flavio Paixao then the diminutive former Clyde and Inverness player.
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