Mikel Arteta's side will head to Paris as underdogs after Ousmane Dembele grabbed PSG's winner in the fourth minute at the Emirates Stadium.
But PSG's first half dominance failed to produced more goals, while Goncalo Ramos and Bradley Barcola wasted two late chances to increase their lead.
The French champions also needed Gianluigi Donnarumma to preserve their lead with fine saves from Gabriel Martinelli and Leandro Trossard.
A disappointed Arteta said in the aftermath that there was still plenty to fight for despite needing to overturn a deficit in the French capital.
"We're disappointed with the result, we put so much into the game," Arteta told Amazon Prime.
"In the first 10-15 minutes we struggled to get momentum and dominate the game the way we wanted; we were very inefficient in where we won the ball and that caused us a few issues, but after that the team were better and better and better.
"It's disappointing not to get a draw at least."
The Spanish coach was impressed by PSG's fast start and said they were fully aware of their opponent's attacking talents.
"That is always the danger, credit to them. They get out of a situation that is close. We had seven players behind the ball and they were clinical.
"Sometimes you just have to recognise the talent of the individual.
"It's half-time and we have a big chance to be in the final."
Arsenal goalkeeper Raya took heart from their battling performance as he looked ahead to the second leg at the Parc des Princes on May 7th.
"They started quick with a goal and they dominated the first 15 or 20 minutes but after that we dominated most of the game, created the chances. Donnarumma made some good saves," Raya said.
"It's only half-time, we take positives. They are a top team but credit to the players for the effort. We could have won the game.
"We know they keep possession a lot and are good at that. They had that chance and put it in the net. We had chances to score but we didn't manage to."
Arsenal failed to score in a home Champions League game for the first time since a 2-0 defeat to Barcelona in the last 16 in 2016.
It was also their first defeat at the Emirates in 18 European matches.
But the north Londoners will have Ghana midfielder Thomas Partey back from suspension for the second leg, which will allow Declan Rice to play a more attacking role.
Raya is convinced Arsenal, who knocked out holders Real Madrid in the quarter-finals, can defy the odds again as they bid to reach their second Champions League final and win the tournament for the first time.
"We showed from the 25th minute of the first half that we can win against any team. We have showed in the season we can win away games too, so we'll go to Paris next week trying to win," he said.
"We have to approach the game the same way. We played well, we put them on the back foot and hurt them. Let's go next week in Paris."
Key insights:
* There were 26 passes in the build-up to PSG's opening goal, the most for them on record in the competition and the most for a goal conceded by Arsenal in the competition on record (since 2003/04).
* Ousmane Dembélé’s opening goal for Paris SG (03:15) was the earliest Arsenal have conceded in the UEFA Champions since Edinson Cavani's goal for the same side in September 2016 (42 seconds), and the earliest they've conceded in the knockout stages of the competition.
* Only Robert Lewandowski (40), Harry Kane (36) and Kylian Mbappé (34) have scored more goals in Europe's big five league in all competitions this season than Paris SG's Ousmane Dembélé (33), while his 25 goals in 2025 is at least five more than any other player in that time.