After claiming the first set, the second-seeded Agassi was up a break in the second but dropped five straight games. At 4-4 in the third set, he fended off three break points before breaking the 19-year-old Youzhny in the 10th game.
"A little drama's always good, isn't it?" asked Agassi, who was playing his first match in three weeks. "It was certainly a lot closer than I wanted it to be, but the first one is always a bit precarious."
Agassi has won five career tournaments on clay, the last coming at the French Open in 1999, when he completed a career Grand Slam.
He is taking part in the Houston for the first time since taking the crown in 1988, when it was held in Charleston, S.C., and has not played in an ATP tournament in Houston in 16 years.
The 31-year-old Agassi missed the first month of the season with a right wrist injury. He returned to reach the final in San Jose in March, and went on to win two of his next three
tournaments, including last month's Tennis Masters Series event in Miami.
Also, seventh-seeded Todd Martin cruised past qualifier Ignacio Hirigoyen of Argentina, 6-0, 6-4, in his first match on clay this season.
In the only other match on Monday's schedule, Russian Nikolay Davydenko routed Lars Burgsmuller of Germany, 6-1, 6-2.
Germany's Tommy Haas, who reached the quarterfinals in Monte Carlo last week, is the top seed this week. Haas was a semifinalist at the Australian Open in January, and advanced to the quarterfinals at Memphis a month later.
The No. 3 seed, Andy Roddick, will try to become the first player to successfully defend his title since Jose-Luis Clerc won consecutive crowns in the early 1980s.
Roddick went 12-1 on clay last season and reached the third round in his first tournament on dirt this year. He won his first title of the year at Memphis and was a semifinalist in San Jose.
Pete Sampras, who suffered a disappointing loss to Alex Corretja in the Davis Cup tie, also in Houston, as well as James Blake and Jan-Michael Gambill and South African Wayne Ferreira fill out the remaining seeds.
The winner earns $51,000.
Celebrity Families of 2014 [PHOTOS]