Following his victory, Djokovic revealed that he was pleased with his response to dropping a set against the Russian.
"It was never going to be an easy match against Mikhail in the quarter-finals of a Grand Slam, and I was aware of his qualities and ability to come back," said Djokovic in a press conference.
"After I lost the third set I definitely tried to regroup and focus on every point individually and start playing the same game that I had in first two sets.
"It was important to get the early break in the fourth. I have done that, and after that I really felt a big relief on the court and just went for my shots and played great."
Djokovic will face Stanislas Wawrinka in the final four.