The England winger's second-half goal was enough to give the reigning champions their second successive win of the new Premier League season and marked Mikel Arteta's 250th league match in charge of Arsenal.
Villa went into the game looking to respond after a difficult start to the campaign, which included a heavy 4-0 defeat against Brighton in their opening match.
Unai Emery's side started brightly and almost took the lead after 12 minutes when Emiliano Buendia struck the crossbar with a powerful effort from distance.
Arsenal gradually began to dominate possession but struggled to create clear chances before the break. Their best opportunity came from a Declan Rice header from a corner, which Villa goalkeeper Zion Suzuki pushed over the crossbar.
Suzuki was making history on his Villa debut as the first Japanese goalkeeper to play in the Premier League, while striker Nicolas Jackson was also handed his first appearance for the club following his club-record 65 million pounds move from Chelsea.
The second half began with Villa again threatening first, as Ian Maatsen fired over from a tight angle.
Maatsen was then involved in another key moment when referee Jarred Gillett initially awarded Arsenal a penalty after Saka was brought down. However, a VAR review showed that the foul had taken place just outside the penalty area.
Martin Odegaard sent the resulting free-kick narrowly wide.
Arsenal finally found the breakthrough just before the hour mark. Riccardo Calafiori made a well-timed run beyond the Villa defence before sending a low ball across goal for Saka, who finished from close range.
The goal opened the match up, with Arsenal looking for a second. Kai Havertz was denied in a one-on-one situation by Suzuki, while Bruno Guimaraes came off the bench to make his Premier League debut for the Gunners.
Arteta's side also handed Ezri Konsa his first Arsenal appearance against his former club after he replaced Cristhian Mosquera.
Arsenal defended their lead well in the closing stages to secure an important clean sheet and continue their perfect start to the season.
Arteta was pleased with the way his side handled a difficult away match.
"Really happy, it's a massive win. It's a tough place to come and earn three points. I think we deserved it," Arteta said.
"We had a great performance. With 11, it wasn't enough, the finishers came in and did a magnificent job to finish the game and win it.
"I know how tough it is to win here. Every big team has struggled against them because they are so good."
The victory was particularly satisfying for Arteta after Arsenal suffered defeat at Villa Park last season.
"Last year was the same. We know how difficult it is to earn points in this league," he said.
"I always trust my players. The level depends on the moment as well. Last year, we lost here in a really difficult way. That was still in the tummy."
Saka, who endured an injury-hit end to last season, has made an impressive start to the new campaign with goals in Arsenal's opening two league matches.
"It's not easy coming here, they made it hard for us. It feels even better to come out with the win," Saka said.
"The way they defend, really compact and with a high line, they caught us offside. We had to adapt to that."
The winger also praised Arsenal's defensive strength after another clean sheet.
"One goal is normally enough for us because we are so solid at the back," he said.
"All the boys put in a shift tonight. It helps with a clean sheet."
Arsenal have now won their opening away league match for a fifth successive season, setting a new club record.
Villa, meanwhile, remain pointless and have yet to register a shot on target in the Premier League this season.
Arsenal's next test comes on Sunday when they host Chelsea at the Emirates Stadium in an early clash between two sides hoping to challenge at the top of the table.