With the Gunners enduring another difficult November having gone into this clash winless in all competitions, Sanchez struck in both halves to seal a 3-1 victory which takes Wenger's side to within three points of leaders Chelsea.
Sanchez opened the scoring when he was gifted possession by a weak Steve Cook backpass before the Cherries - without the ineligible Jack Wilshere who is on loan from Arsenal - levelled after Callum Wilson converted a penalty he won after being felled by Nacho Monreal.
The hosts came out stronger after the break and re-established their lead through Theo Walcott's header as the England international celebrated the birth of his second son in style.
Sanchez then made sure of the points by sweeping home after a fluid move with the game just heading into stoppage time.
The 27-year-old now has 10 goals for the season since his move to a central attacking role, and Wenger was once again left praising the former Barcelona man.
"Even when he looks dead, he's still alive and always finds the resources to do something special," he said.
"He has that conditioning which allows him to always find an extra gear to be dangerous.
"Bournemouth are a tricky side who can keep the ball and, if you're not well organised, you can be in trouble.
"On top of that, I felt we started well but when they equalised we were nervous and that's where having three consecutive draws played a little bit on our minds."
While Hector Bellerin and third-choice goalkeeper Emiliano Martinez penned fresh contracts at the Emirates Stadium in the last week, it is new deals for key duo Sanchez and Mesut Ozil which Arsenal fans are waiting on.
And Wenger said he knows that tying Sanchez down for longer is key, even if performances like this mean he could demand a bigger pay-packet.
Asked about contract discussions with the Chile forward, the Arsenal boss replied: "It is important
How important it is, I don't know
We will not sort it tonight
We have some time in front of us
You (the media) like us to spend money, so you should be happy!"
There was one low point for Arsenal as Mathieu Debuchy managed just 15 minutes on his comeback game before being forced off with a hamstring problem.
The France international last started for Arsenal 370 days ago and was a surprise name on Wenger's teamsheet with Carl Jenkinson having filled in for the injured Bellerin of late.
And Debuchy, who spent the second half of last season on loan at Bordeaux having attracted the attention of Manchester United, has told Wenger he thinks it could be a serious injury.
"Debuchy, he thinks it's a severe one," he added.
"Don't go too far because I have to speak to medical people, but you can only assess it really after 48 hours
You have to let it bleed and after, usually they have an MRI 48 hours later to see how big the damage is.
"When I picked the squad I imagined that something could happen to Debuchy because he's had many setbacks."
Bournemouth boss Eddie Howe had seen his side win at Stoke last weekend and he felt the Cherries deserved more in this outing.
"I thought it was harsh on us today, I thought we were well in that game," he said.
"We gave them the first goal and got a number of chances we didn't take
The 3-1 scoreline is difficult for us to swallow today.
"We have been unlucky, we feel there are a couple of incidents I believe have gone against us, I think every team will say that at the end of the season.
"I thought we were very good today
It is always difficult when you go 1-0 down early on in the game, especially in that manner - an individual mistake."
Source : PA
Source: PA