The Gunners, who had Ainsley Maitland-Niles sent off in the first half, looked to be heading for a sorry home defeat as John McGinn and Wesley scored for Villa either side of Nicolas Pepe's first goal for the club.
Calum Chambers levelled in the 81st minute, though, and Pierre-Emerick Aubameyang sealed a dramatic three points with a superb free-kick six minutes from time.
That late jubilation was a far cry from the scenes that greeted Xhaka's removal for Joe Willock earlier in the half, with large numbers of the home faithful audibly expressing their dissatisfaction with the Switzerland international's display.
Emery, however, said the 26-year-old Xhaka is still very much part of his plans and wants to see more support for him from fans.
"We need our supporters and we need them helping each player," he told a media conference. "It is our responsibility to transmit our commitment and Xhaka is doing this.
"He played very well on Thursday [against Eintracht Frankfurt in the Europa League] and last week at Watford he did well. Here, I changed him because he played for 90 minutes on Thursday and I wanted to bring fresh players on.
"But Xhaka is a very important player for us. He has a big commitment here, he is improving. I am not asking people to support him but I assure you he is going to achieve his best things if the supporters help him.
"We are professional; Xhaka is an experienced player and he knows sometimes we can receive criticism and we need to be mature and continue working.
"For me he is a very important player and I am going to support him. I am going to decide when he plays and when he doesn't, but he is important for us because his commitment and behaviour is great and he is a good player."
Matteo Guendouzi was central to Arsenal's stirring second-half display, winning the penalty which was converted by Pepe and driving forward the Gunners at every opportunity.
Emery was full of praise for the 20-year-old and said he has exactly the kind of attitude he wants from his players.
"With Matteo, last year when he came here he progressed and he is progressing," he said. "Really, I like that spirit he has. He has talent but above all I like his attitude, his energy, his commitment and his behaviour.
"He's one player who is angry when he is not playing but I like that. When you decide that they play you know they will give you what you need.
"I want to be calm with him. He is humble, continuing to improve and growing up with us."