Liverpool created the first clear sight of goal after 13 minutes when Trent Alexander-Arnold found space down the right flank, playing a low pass into the box which Daniel Sturridge turned goalwards, only to be denied by a fine block from Scott Dann.
Sturridge turned provider shortly afterwards when he flicked Jordan Henderson's lofted pass into the path of Adam Lallana, but the midfielder could not keep his half-volley down and blazed it over the crossbar.
Mohamed Salah was the only new signing on either side to be included in the starting lineups, and he almost made it two goals in two games for the Reds midway through the half when he cut inside onto his left foot and curled an effort narrowly over.
The match saw new Palace boss Frank de Boer get his first look at many of his first-team players, but it took them until the 27th minute to test Simon Mignolet for the first time when Keshi Anderson exchanged passes with Andros Townsend before being denied by the Liverpool keeper.
The Palace keeper was forced into a more testing save with only two minutes remaining before half time when Sturridge cut inside from the left flank and drilled a low strike towards the bottom corner, which needed a smart one-handed stop from Hennessey.
Lallana came close again moments later when his deflected shot flew wide of the target, but despite seeing the majority of the ball in the opening 45 minutes Liverpool went into the break still goalless.
Sturridge showed signs of rust early in the second half when a poor touch and tame finish saw a decent chance go begging, and that proved to be his last action as he and every other Liverpool outfield player was subbed off in the opening stages of the half.
Among those to come on was Philippe Coutinho, who almost delighted the Liverpool-heavy crowd shortly after his introduction when he cut inside before curling an effort narrowly over the crossbar.
It was another one of the new faces that finally broke the deadlock just after the hour mark, though, as Solanke opened his Liverpool account with a powerful and clinical finish into the bottom corner from outside the area.
Palace introduced new loan signing Ruben Loftus-Cheek among their second-half subs, and the Eagles had the chance to level things up with a rare attack 15 minutes from time when Patrick van Aanholt's cross found Aaron Wan-Bissaka at the back post, but the youngster failed to catch his ambitious attempt at a volley and the opportunity went begging.
Liverpool put the game beyond doubt with just over 10 minutes remaining when Coutinho's blocked shot fell into the path of Origi, who applied the finish from close range.
Ryan Kent almost added a third late on for the Reds when he cut inside from the right flank before drilling a low shot towards goal, but Hennessey got down to turn the shot around the post and the resulting corner eventually ended with Coutinho seeing a tame shot saved.
Liverpool will now face Leicester in Saturday's final, before which Palace will take on West Bromwich Albion in the competition's third-place playoff.