Bournemouth had their moments on their visit to North London but they were made to pay for missed chances as Arsenal moved back to within three points of Premier League leaders Chelsea.
The Cherries player attempted to play the ball back to goalkeeper Adam Federici, but it lacked pace and Sanchez was able to step in and slide the ball into the corner from 18 yards.
However, the Gunners were soon dealt a blow when Mathieu Debuchy - who was making his first Arsenal appearance in over a year - was forced off through injury and it was not long before the home team were pegged back.
The equaliser led to Bournemouth pushing for a second and they should have had one through Adam Smith, who headed over from eight yards, and Brad Smith who sent the ball too close to Cech at the near post.
It took time for Arsenal to respond to the lively response from their opponents, but they almost regained their advantage in first-half stoppage time when Sanchez struck the bar with a shot from 16 yards, with Federici appearing to get the smallest of touches to divert it onto the woodwork.
The Gunners returned for the second half on the front foot and it took them just eight minutes to regain the lead as Walcott was perfectly placed at the back post to head Monreal's cross into the net from four yards out.
However, Arsenal failed to build on their lead and they were perhaps fortunate not to have Sanchez sent off for a wild lunge on Harry Arter, with the Chilean's studs going down the back of his opponent's Achilles.
Arsene Wenger introduced both Aaron Ramsey and Olivier Giroud, but their arrival was almost met by Bournemouth netting a second goal as Cech pulled off an excellent save to deny Benik Afobe from close range.
Substitute Lys Mousset also headed wide for Bournemouth in the final minute of normal time before Arsenal made sure of the three points when Sanchez was on hand to tap into an empty net after fine work from Giroud.
It was a goal that gave Arsenal a first win in four games, while they extend their unbeaten record in all competitions to 19 games.