The 40-year-old Argentine takes over from Eder Sarabia, who left the club after guiding Elche to promotion to La Liga and securing their survival in the top flight, before deciding to step away from management for personal reasons.
Anselmi will take on his first role in Spanish football, having built his managerial reputation elsewhere before this appointment.
He is set to spend his early days at the club familiarising himself with the training facilities and the Martinez Valero stadium, ahead of a formal media presentation expected later this week.
The Argentine coach will work closely with club owner Christian Bragarnik, general director Pedro Schinocca and sporting director Pepe Contreras as Elche finalise their squad planning and prepare for the pre-season schedule.
Sarabia's exit leaves a significant void at the club after a hugely successful campaign, with Elche securing both promotion and survival in the same season under his guidance.
Anselmi's task will be to build on that platform while adapting his methods to a club and a competition entirely new to him.
The early stages of his tenure are expected to focus on assessing the existing squad before any incoming or outgoing transfer activity begins in earnest.
Elche's hierarchy will be hoping the appointment provides the continuity needed to consolidate their place in La Liga after an eventful rise through the divisions.