Mikel Arteta's side remain in 10th position in the league but could end up in the bottom half if results don't go their way.
Villa recovered from their midweek loss against West Ham United and are five points off the top four places with a game in hand.
"I'm extremely disappointed to lose and the way we lost it. We were the better team," Arteta told BBC Sport.
"We controlled every department but we gave them the ball. We gave them another three big chances. Then when we had to score the goal, we didn't find the right pass or the right shot.
"We created enough opportunities. We didn't score, we cannot win the game."
Villa broke the deadlock inside the opening two minutes after some poor Arsenal defending which resulted in Watkins' deflected strike going into the back of the net.
Partey limped off after the hour mark and Jack Grealish almost doubled Villa's lead but Ryan produced a fine save to deny him.
Arsenal almost equalised late on but Martin Odegaard's close range shot blazed over the bar.
Watkins also saw a strike go just past the post in the latter stages of the game but Villa survived to secure an important victory.
"The values of being a good team, the work ethic... I thought we saw that in bundles today against a really good offensive team," Villa manager Dean Smith said.
"We started with a good intensity that set us up really well. It had to be a good defensive performance but their keeper had to make some really good saves as well."