After the intensity of Wednesday's 0-0 Champions League draw away to Atalanta, the last thing Celtic wanted was Motherwell to fly out the traps - but that's just what they did.
Teen sensation Miller hit the post with goalkeeper Kasper Schmeichel in no man's land, before curling a free-kick on to the crossbar two minutes later.
However, Celtic soon matched their hosts' tempo, creating plenty down the right with James Forrest and Johnston on fine form - but it was the deputising McCowan who broke the deadlock.
Taking one touch to get away from Kofi Balmer, he was not closed down and from the edge of the box managed to roll the ball into the bottom corner.
That settled any nerves that Motherwell had created and the hosts struggled to really threaten Celtic's goal again.
Some courageous home defending, and some decent stops from Aston Oxborough, meant the second goal did not come until the hour mark when Alex Valle's brilliant cross was met by Johnston and the full-back guided his header in.
The strength of Celtic's bench was notable and it was two subs who combined for the third. Idah, who had already hit the post, swept in a Kuhn cutback and the game was sealed.
Motherwell's misery was further compounded when Gordon was sent off following a VAR check in added time. The home skipper could have no complaints - his studs-up tackle on Idah merited red.