The Belgian - widely criticised after the previous week's draw with Aberdeen - produced a first-half penalty save from Marius Niculae.
Hibs had the prefect start when referee Eddie Smith awarded them a penalty after just 85 seconds when a wayward header from David Murphy struck Stuart McCaffrey's hand.
Fletcher made an expert job of the spot kick - sending Mike Fraser the wrong way as he placed the ball to the keeper's right.
It was obvious from the outset it was going to be an open game with Hibs looking jittery at the back in the early stages.
Makaba-Makalamby took no chances as he palmed a Barry Wilson corner over his own crossbar as Inverness won a string of early corners.
Ross Tokely headed wide before Wilson drilled in a 20-yard shot which presented no problems for Makaba-Makalamby.
Craig Brewster came closest when he volleyed just a yard wide after Roy McBain and Ian Black had combined to set him up.
Hibs hit straight back and a flowing move between Lewis Stevenson and Fletcher set up a chance for Abdessalam Benjelloun, but he slipped the ball wide from six yards with Grant Munro doing enough to put him off.
Fletcher then looped a header over the top from a Benjelloun cross but Fraser had it covered.
If anything, Caley Thistle had more of the ball in the first half but too often wasted their possession by selecting the wrong final option.
Hibs always threatened on the break. Zemmama played in Benjelloun but he seemed to stub his toe as he shot and the ball rolled harmlessly through to Fraser.
But Caley Thistle should have levelled seven minutes from the break. Rob Jones was adjudged to have pulled Brewster down in the area and the Easter Road skipper was booked for his protests.
Niculae struck his penalty well enough but Makaba-Makalamby managed to stick out a leg to deflect it over his crossbar for a corner.
Hibs almost made Caley Thistle pay immediately as they raced up the pitch in the best move of the match but Fletcher's final shot was beaten out by Fraser.
Both sides had chances in the second half. Fletcher shot narrowly over and then Brian Kerr slipped a shot wide.
The closest Caley Thistle came to an equaliser was nine minutes from time when Brewster's deflected shot looped narrowly wide.