A must-win game for Brendan Rodgers' men looked to be slipping away when Blair Spittal's sensational strike from the edge of the area found the top corner just before half-time.
Idah replaced Kyogo Furuhashi at the break and the Irish international made a quick impact as he towered above the Motherwell defence to head into the far corner.
Chances then came and went for the visitors as Daizen Maeda twice failed to hit the target with the goal gaping.
Rangers were rampant on Saturday with a 5-0 thrashing of third-placed Hearts to pull five points clear.
But Celtic stayed in the title race as on-loan Norwich striker Idah slotted home Alistair Johnston's cross.
The Hoops then sealed the points in the 96th minute as another two substitutes combined when Luis Palma tapped in Yang Hyun-jun's cross.
"It's a very important win. I think the players responded magnificently second half," said Rodgers.
"The key thing for us is to stay together. It's still a long way out, even if we draw the game today.
"Four points isn't a lot in my experience. Four points would have been nothing, but it was important for us to get the win today psychologically."
Defeat leaves Motherwell still just four points above the relegation playoff place and six adrift of a place in the top six.
