Rodriguez and fellow Border Patrol agent Ricardo Guillermo Salinas were among four people killed Tuesday by Moore in what apparently started as a jealous rampage over an ex-girlfriend.
It ended with Moore ambushing law officers who were searching for him and dying in the ensuing shootout.
"We in INS and in this community have suffered a tragic loss," said Doris Meissner, Immigration and Naturalization Service commissioner.
Vicki Buffolino of CBS affiliate KENS reports the rampage started when police in the Harlingen suburb of Rio Hondo received a 911 call of shots fired at a house at about 5:30 a.m.
An officer found 53-year-old Margarita Flores and her daughter, 31-year-old Delia Morin, shot to death in a hallway in their home. Flores' son, Dan Morin, 22, was critically wounded.
Rio Hondo Police Chief Carlos Gonzalez said Moore went to the house seeking a former girlfriend who was a house guest. She was not injured.
Newspaper reports said Dan Morin had been dating Moore's ex-girlfriend, Julie Cox, for about a month and Morin demanded Cox leave with him when Flores unexpectedly emerged from a bathroom, and Moore turned and started shooting.
Moore fled in a pickup truck. Cameron County Sheriff Omar Lucio said deputies spotted the truck and gave chase. Other officers spotted the vehicle near the Moore family home, about 10 miles away near San Benito.
A group of officers, including three Border Patrol agents who joined the pursuit at the request of local authorities, went to the Moore home to talk to his parents.
When they were preparing to leave, Moore suddenly emerged from a nearby cornfield and opened fire with a semiautomatic rifle, the Texas Department of Public Safety said.
"He actually ambushed the officers and he opened fire,"Lucio said.
"I jumped out of bed and I looked out, " neighbor Lucy Preco told KENS. She said she saw only silhouettes coming around the house.
Rodriguez, 28, was the daughter of retired El Paso Deputy Border Patrol Chief Steve Williams. She was the first female Border Patrol agent killed in the line of duty.
Moore, 25, was the son of a longtime Harlingen police officer and the brother of a San Benito officer. His family was well-known in the city.
Rodriguez, the mother of a 1-year-old girl, was married to an immigration officer. She joined the Border Patrol in 1992 and had been assigned to the Harlingen station ever since.
Salinas, 24, who was from San Antonio, had graduated from the Border Patrol academy last January.
"The deaths of these fine, young people cannot go unpunished,"Meissner said of the INS.
In the 74-year history of the Border Patrol, 81 agents have been killed in the line of duty, including two previou deaths this year.