The goalkeeper fumbled Joleon Lescott's second-half header into his own net as Remi Garde claimed his first victory at the 11th attempt.
It was also redemption for Lescott, who was accused of playing a game with chewing gum - to the fury of fans - while a substitute during Saturday's 1-1 FA Cup draw at Wycombe.
And the defender gave Villa much to chew over, even if their first home league win of the campaign failed to lift them off the bottom of the table.
It closed the gap to safety to eight points and was just their second league victory of the season, coming 156 days after the first, but it was deserved after a brave performance.
The claret and blues require a monumental effort to survive but, on this evidence, at least they will go down fighting.
Wilfried Zaha hit a post for Palace inside the first minute, but the Eagles deserved nothing from a limp performance.
Libor Kozak made his first league start for Villa since Boxing Day 2013 - when he also faced Palace - while returning goalkeeper Mark Bunn's last league start came against Villa for Norwich in May 2013.
That Villa were now relying on striker Kozak - whom Garde had stated could leave this month - showed the depth of their problems.
And they almost got worse after just 44 seconds as Villa survived a massive scare.
They were carved apart on the right as Jason Puncheon and Zaha swapped passes to leave Zaha free in the area and he curled a shot against the outside of a post.
But Palace never got as close again as the let-off woke Villa up and they tried to force the issue without breaking the visitors down
Without a recognised front man Palace were still dangerous on the break as Zaha, Puncheon and Bakary Sako - one of three Palace changes - threatened to find the gaps in Villa's creaking defence.
But it was Damien Delaney who came the closest after half-an-hour when he directed Yohan Cabaye's free-kick inches wide.
Jordan Ayew had already hit the side-netting with Villa's best opening, but they were too tempted to shoot from distance, often wasting any openings.
Palace brought on Connor Wickham to spearhead their attack at the break, but they almost fell behind six minutes later.
Hennessey had little to do previously, but he was forced to turn Kozak's deflected effort wide as Villa threatened to draw first blood.
And the goalkeeper then handed them the winner on 58 minutes with a monumental error.
Villa had been pressing and won a corner and, despite Lescott's header flying straight at Hennessey, he spilled it through his legs and over the line.
The hosts deserved the lead and Alan Pardew threw on Marouane Chamakh in response, but Palace lacked punch.
The Eagles had the third best away record in the division coming into the game, but offered little evidence of their prowess on the road.
Ayew fired over for Villa before Scott Dann headed well over with four minutes left.
Surprisingly, though, there were few nerves in the final moments, even as Dann missed his kick in stoppage time as Villa hung on.
Source : PA