Defeat ended Liverpool skipper Steven Gerrard's dream of bringing his 17-year career at Anfield to a close with a cup winner's medal on his 35th birthday and Villa will play Arsenal in the final on May 30.
Liverpool made the breakthrough in the 30th minute when Philippe Coutinho scored after sloppy Villa defending.
But Villa were only behind for six minutes before they deservedly equalised when Delph, guilty for his part in the build-up to the Liverpool goal, did everything right.
The Villa skipper surged forward before playing a one-two with Jack Grealish and crossing for Benteke whose first time shot flew past Simon Mignolet into the net.
Villa went ahead nine minutes into the second half when Delph scored after a superb move which included a Benteke backheel and a Grealish through ball to Delph who cut in and gave Mignolet no chance from 12 metres.
The opening spell was cagey with Villa, rejuvenated in the last two months since Tim Sherwood became manager, more than holding their own against Brendan Rodgers's team who welcomed Gerrard back after a three-match suspension.