He described it as one of the deadliest days in the state's history.
The massacre happened at a Walmart store on Saturday near the Cielo Vista Mall, a few miles from the US-Mexico border, the BBC reported.
A 21-year-old man is in custody and is believed to be the sole gunman. Mr Abbott praised the police officers who apprehended him.
The suspect has been named by the US media as Patrick Crusius, a resident of the Dallas area.
CCTV images said to be of the gunman and broadcast on US media show a man in a dark T-shirt wearing ear protectors and brandishing an assault-style rifle.
The first reports of a shooting emerged at about 10 a.m.
Police said the Walmart was "at capacity" with shoppers buying back-to-school supplies at the time of the attack.
Police spokesman Sergeant Robert Gomez said the only suspect was a white male in his 20s. No officers had fired their weapons in apprehending him, he added.
"We have secured Walmart and we have secured Cielo Vista Mall. We don't feel that there is a threat to the public or that there are any other shooters at this time," Sgt Gomez said.
El Paso Police Department tweeted that blood donations were "needed urgently".
A man who was in the shopping centre at the time of the attack recalled the moment that he found out about the shooting.
"There were employees, they were coming in and they would tell me that they heard some gunshots and that people just started bolting straight into the store in order to get to cover. In the meantime I was just trying to stay calm, but I was freaking out internally," he told a media outlet.
Another witness, Glendon Oakly, said he was in a sporting goods store inside the shopping centre when a child ran inside "telling us there's an active shooter at Walmart".
Oakly said no-one took the child's claim seriously, but just minutes later he heard two gunshots.
"I just thought about getting the kids out of the way," he said.
( With inputs from IANS )