The Spanish side will take a slender lead into the second leg of the Europa League semi-final next week, though the margin between the teams could have been far greater at the halfway stage of the contest.
Manu Trigueros and Raul Albiol scored as Unai Emery's side threatened to overrun his former employers in the first half, yet the Gunners came out a different side after the interval.
Despite the dismissal of Dani Ceballos, Bukayo Saka won a penalty that Nicolas Pepe converted to cut the deficit in the 73rd minute.
Villarreal then had Etienne Capoue dismissed before the returning Pierre-Emerick Aubameyang had a chance to equalise in added time, only to be denied by home goalkeeper Geronimo Rulli.
"We didn't want to come here and lose, but after the way the game developed, and being 2-0 down with 10 men, you have to take the result," Arteta told BT Sport.
"If you have to lose, this is the best result that we could have. But two different halves."
Asked what had led to the change in his team after the break, he said: "We started to be us. In the first half, there were so many moments when we weren't us.
"We were disorganised, we started to chase, were not precise enough with the ball nor occupying the right spaces, we didn't have enough control.
"We didn't carry enough threat, or have the right desire to attack the opponents' box. In the second half it was completely different."
Arsenal was indebted to a fine save from goalkeeper Bernd Leno when the score was at 2-0, the German denying Gerard Moreno.
So, while Emery became the first ex-Arsenal boss to beat the Gunners since George Graham's Tottenham side won 2-1 at White Hart Lane in November 1999, the tie is firmly in the balance ahead of the second leg at Emirates Stadium.
Villarreal has progressed from 15 of their 16 two-legged ties in major European competitions when winning the first leg – the only exception coming in the 2015-16 Europa League semi-final, when they went out to Liverpool despite winning the first meeting.
On Leno's save from Moreno, Arteta said: "He was very good when we needed him at 2-0. The chance with Gerard is probably the biggest one they had.
"He saved us, but then at the end we had a big chance with Auba and we weren't able to get it in."