Kilmarnock 0 Hibernian 2
The visitors struggled in the first half, but a bold half-time change from John Collins made the difference to the result.
Killie's Gary Wales came closest to breaking the first-half stalemate after excellent interplay with Danny Invincibile, but the former Hearts striker was unlucky to see his 20-yard shot rebound off the far post and fall back into the goalkeeper's arms.
In what was anything but a dirty match Charlie Richmond brandished eight yellow cards and one red.
From the 16th free-kick awarded to Hibs in the first half, they had their best chance. Rob Jones won the first header and Momo Sylla failed to clear leaving Dean Shiels with a clear shot from seven yards, but he blazed over.
Hibs boss Collins was clearly unimpressed by his team's first-half performance and made a double half-time change, taking off Kevin Thomson and Ross Campbell and introducing Guillaume Beuzelin and Ivan Sproule.
The change brought instant rewards as the subs combined to give Hibs the lead within five minutes. Beuzelin slid the perfect pass for Sproule to rifle home from the edge of the area.
Sproule ran past the Killie defence on the hour mark before firing in another shot. This time Alan Combe parried and Steven Fletcher should have nodded home, but headed tamely over.
Killie rarely threatened in the second half, but when they did, Steven Naismith was involved. His driving run and shot in the 72nd minute grazed the post.
The home side's task was made even harder when Wales was shown two yellow cards in the space of three minutes to take an early bath.
Hibs ensured the victory four minutes from time after classic wing play from young Lewis Stevenson, who cut the ball back for Fletcher to sweep the ball home.