The Scot took control of the first set in the early stages with a break of the Youzhny serve before holding to move 3-1 ahead in the opener.
The Russian did have the chance to break back at 3-4, but Murray held firm to move into a one-set lead with a powerful forehand.
Another break in the early stages of the second allowed Murray to take control, but Youzhny responded with consecutive breaks of his own to go 5-2 ahead in the set.
Murray refused to be beaten, however, and recovered to send the second set into a tie-break, which he eventually took 7-5 to move into a comfortable lead.
It proved to be more straightforward for the Brit in the third and what proved to be final set, with two breaks and four holds booking his spot in the final eight on Centre Court.