Edmund, the fifth seed at his home Grand Slam, appeared to be in big trouble as he lost the opening set and proceeded to be broken at the start of the second.
But eight games in succession from the Brit saw him roar back to seize control of the match as the 15-year-old Kozlov – three years his junior – faded under the pressure.
The American saw his serve broken at the start of the deciding set and suffered an injury after slipping a further break behind Edmund, who by this point was being roared on by a packed crowd on court three.
Edmund took the final set in 48 minutes to wrap up the match, after having taken the second in just 24 to get himself back into the contest as he sealed his progression through to the last four.
The Brit will next take on Gianluigi Quinzi of Italy - the sixth seed - in the semi-finals at SW19. The other last-four clash sees Hyeon Chung take on Germany's Maximilian Marterer.