The British number two began the week at a career-high world ranking of 43 and he has taken a step closer to moving further up the standings with a 7-6 7-6 win over the Ukrainian.
Both players broke once during the opening set but it was Edmund who was able to edge the tie-break 7-5 to move closer to the last 16.
Marchenko held a 5-2 lead in the second before Edmund won the next three games and squandered five opportunities to serve for the match.
The 21-year-old then had to save two set points to prevent Marchenko taking the contest into a decider but he held his nerve during the second tie-break to again take it 7-5.
Edmund will now face David Ferrer for a place in the quarter-finals.