The Serb, seeded 20th after a well-documented form slump, was given a ferocious fight in the early stages with the opening three games lasting 29 minutes on Court Philippe Chatrier.
But what looked like being a long evening for the 12-times Grand Slam champion ended up being a reasonably comfortable one as he kept Verdasco at arm's length.
One break of serve was enough to take the opening set and although left-hander Verdasco produced thunderous winners to stay in touch in the second his challenge soon faded.
After arriving in Paris with his lowest seeding since 2006 and his expectations not much higher, the former world number one appears to be rediscovering his mojo at the right time.
The draw has fallen in his favour with Verdasco having removed fourth seed Grigor Dimitrov and he had another slice of fortune on Sunday when eighth seed David Goffin was beaten by unheralded Italian Marco Cecchinato.
Cecchinato will hold few fears for Djokovic who moved up a gear against Verdasco - a dangerous opponent who has three claycourt victories over the Serb.
Verdasco saved a break point in a 22-point second game and exerted huge pressure on Djokovic in a marathon third game but could not convert two break points of his own.
The pace was getting to Verdasco and he hit a forehand long in the next game to give Djokovic a decisive break.
Verdasco recovered a break in the second set, forcing Djokovic to net a stretching volley.
Djokovic replied immediately, finding the baseline with a deep drive that forced a miss-hit from the Spaniard.
Verdasco's hopes of reaching his first French Open quarter-final were sunk when Djokovic nailed down a break at the start of the third set before cruising to victory.
No other player has reached 12 French Open quarter-finals in the professional era, although 10-times champion Rafael Nadal can match Djokovic's feat if he wins on Monday.
