After 10 minutes, Scott Brown saw his shot saved by Randolph, before Victor Wanyama pounced on the loose ball, but he couldn't find the target.
Kris Commons fired a shot just over as the Bhoys continued to pile the pressure on Motherwell.
With 14 minutes gone, Commons was involved again when he was brought down as he got in on goal. The midfielder could not take advantage of the resulting free kick, hitting it against the wall.
Celtic finally made it 1-0 on 32 minutes, when Commons passed to Hooper, who showed perfect control before firing beyond Randolph.
The away side made it 2-0 on 34 minutes. Brown found himself in behind the home side's defence and saw his cut-back inadvertently put into the net by Motherwell's Adam Cummins.
Ten minutes after the restart, Celtic were awarded a penalty when Cummins hauled down Commons in the area. The defender was sent off by the referee for the foul.
Brown took the penalty, but his low effort was saved by Randolph.
A cross from Emilio Izaguirre was met by Brown, but Randolph was equal to the effort once more.
Motherwell could not get into the game as Celtic lay siege on their goal. Wanyama and Izaguirre came close, but they could not find a way through.
The final whistle blew with the score at 2-0.