Celtic 2 Kilmarnock 0
Darren O'Dea had put the home side ahead shortly before half-time, but it was not until Aiden McGeady added a second in the final minute that the home side could relax.
In fairness to Celtic they also hit the woodwork three times in the second half, though Kilmarnock had one effort against the bar.
Rhian Dodds shot straight at Artur Boruc in an early chance for the visitors before Craig Beattie pulled his shot wide after a great pass from Derek Riordan.
In a swift counter-attack, Kilmarnock outnumbered the Celtic defence. Garry Hay's pass put Dodds through on the right but the winger shot inches wide.
Celtic punished Kilmarnock for the miss by taking the lead shortly after. Shunsuke Nakamura's free-kick fell to O'Dea who hooked the ball home from six yards.
On the hour mark Riordan's shot from the edge of the box came back off the bar before Maciej Zurawski drilled a shot straight at Graeme Smith.
Kilmarnock manager Jim Jefferies was sent to the stand after an altercation with the linesman who had infuriated him when awarding the free-kick which lead to Celtic's opening goal.
Danny Invincibile missed a good chance after the beating the offside trap before Fraser Wright headed Peter Leven's free-kick against the bar.
This was as close as Kilmarnock came to equalising, and after Jiri Jarosik and substitute Jan Vennegoor of Hesselink had both seen efforts come back off the woodwork, Celtic scored their second in the last minute.
From Nakamura's flicked pass McGeady rolled his shot under keeper Smith.