Alexander, 49, led the club to promotion to the English Football League in his first season in 2018-19.
He leaves the club in fifth place in League Two, having gone unbeaten in their first five games this season.
Scholes' only previous experience in management was 31 days as boss of hometown club Oldham Athletic in 2019.
He resigned from his position as he was "not able to operate as I intended".
His first game as Salford boss will be Saturday's trip to Port Vale.