The incident occurred in the second half, when the game was still goalless. Cahill was booked for a rash challenge but as Kompany took evasive action, he landed on the midfielder's leg.
It quickly became apparent Cahill would not be able to continue and he was replaced by Louis Saha. Moyes reported afterwards his player had only suffered bruising to his shin but felt the damage could have been far worse and pointed the finger at Kompany.
"The boy does him," Moyes told the BBC.
"He definitely sees his shin going in and stands on it. People who have been players know what I am talking about.
"I don't think it is too bad. It just looks like bruising. But they are the ones where it doesn't take much to crack your shin. The officials seemed to spot most other things. But they did not spot the big one. Referees are judged on big decisions."
In fact, Moyes was not impressed by the performance of World Cup final referee Howard Webb and his officiating team, whom he felt showed inconsistency in the way they handled the game and got a key decision wrong in the build-up to Mario Balotelli's opener.
"Overall, we found the way the officials treated us really difficult to take," said Moyes.
"We found it difficult to talk to them and found the fourth official was very much the same."
Everton finished the game with five yellow cards compared to City's one, which Moyes did not feel was a fair reflection of how the game panned out, pointing out one contrast where an Everton player did not get a free-kick for a challenge very similar to one Leon Osman was booked for earlier in the game.
"There is a challenge on the edge of the box by Vincent Kompany on Louis Saha that was not much different to the one by Leon Osman on Micah Richards," he said.
"What we want is a bit of consistency."
Moyes' frustration at failing to secure a point was obvious.
After all, against opponents he has been used to beating down the years despite the vast disparity in finances, he came up with a game-plan that looked like paying off until the introduction of Balotelli after an hour.
It did not take long for the Italian to bag his first Premier League goal since February and set City on a victory march that was secured by fellow substitute James Milner in the final minute.
"I was lucky with the substitutions," said City manager Roberto Mancini.
"He (Balotelli) has worked well in the last few weeks but hasn't played. This time he deserved to play for half an hour and decided the game."
The other factor behind Mancini's decision, which meant Carlos Tevez remained on the bench for the entire duration of the match, was Balotelli's four-game European ban, which means he will sit out Tuesday's trip to Bayern Munich.
Under such circumstances in the past, Balotelli might have maintained his pout, even after scoring.
Not on Saturday. Instead he raced across the pitch to give his manager a hug, even though Mancini would understand if the temperamental 21-year-old had been upset at being left out once more.
"I want to see this reaction," he said.
"I am the manager. I can only play with 11 players and sometimes you can't play. It is important that when I call these players, they are ready."
City are now level on points with neighbours Manchester United, trailing only on goal difference. And it seems they have also learned the art of grinding out wins rather than merely producing the free-flowing football they evidently have the players for.
"In this game you might have the ball all the time, create a lot of chances, then lose on the counter-attack," said Mancini.
"This time, we deserved to win."