Agut, World No.14 and second seed, needed just one hour and 15 minutes to quell a spirited challenge from Brazil's Rogerio Dutra Silva 6-3, 6-2 at the SDAT Stadium.
The seventh seeded Youzhny was equally imperious, romping past Argentina's Renzo Olivo 6-1, 7-5.
"He is a former Top 10 player and a tough opponent," Agut said, speaking of their upcoming face-off.
"But I will make him work on the court and he will have to fight it out if he wants to beat me," he added.
Agut didn't show any signs of rustiness despite playing his first match after a two-month break. He dictated the proceedings by serving hard and unleashing strong backhand and forehand winners.
He broke Dutra Silva thrice, in the sixth game in the first set and the fifth and seventh games in the second set.
The 28-year-old Spaniard was pretty happy about his game.
"I felt great on the court, without any match practice," Agut said, looking ahead to the new year.
"It was a little windy. But the conditions are good, the court is good, to play at night," he added.
World No.57 Youzhny and Olivo traded breaks in the first three games before the former took charge of the match.
He broke Olivo in the fifth and seventh games to seal the first set 6-1 under 30 minutes.
The second set went all the way till the wire before Olivo, serving at 5-6 to save the match, double-faulted to hand the match to Youzhny.