Anichebe had come off the bench to net a last gasp winner against West Brom at the Hawthorns on New Year's Day after a lengthy injury lay-off.
Villa striker Darren Bent's seventh goal of the campaign had broken the deadlock but the Toffees deserved their equaliser.
Alex McLeish's side at least ended their run of four successive home defeats with Stephen Ireland having one of his best games in midfield.
But their star performer was recalled keeper Shay Given who had been sidelined for six weeks with a hamstring injury.
He justified his instant return to action in place of Brad Guzan with several fine saves.
Everton were well worthy of a share of the spoils with Landon Donovan impressing on the right of midfield.
Villa were the first to threaten after Bent made a probing run towards the Everton box and was brought down by John Heitinga.
Villa skipper Stiliyan Petrov's 25 yard free-kick contained plenty of power and was fisted away by the diving Tim Howard.
Villa were the more purposeful in the early stages and Marc Albrighton's shot was blocked after Alan Hutton had teed up an opportunity.
Everton came more into the game and new signing Darron Gibson's flighted pass picked out a run into the Villa area from Tim Cahill but his first touch let him down.
Given made an excellent save away to his left to keep out a Louis Saha glancing header from a Leighton Baines free-kick.
He then had to turn aside a misdirected header from his own defender Stephen Warnock from a Royston Drenthe centre.
Drenthe made a crucial interception at the far post, tracking back inside his own area to deny Albrighton following a cross from Gabriel Agbonlahor.
Everton appealed in vain for a penalty after Hutton appeared to pull back Saha as he attacked a Donovan centre.
Marouane Fellaini became the first player to be yellow carded after 35 minutes for bringing down Stephen Ireland after he had dispossessed the Everton player.
Villa finished the first half strongly and Bent headed over from a Warnock centre four and Howard parried a curling effort from Agbonlahor around the post.
Given made his third fine save of the game at the start of the second period after Collins brought down Drenthe.
Baines' curling free-kick found Fellaini unmarked at the far post and his powerful header was blocked by the Republic of Ireland keeper.
Howard parried a shot over his shoulder from Bent before the Villa striker broke the deadlock after 56 minutes.
Albrighton's centre was only half cleared and Ireland poked the ball across the box for Bent to net from close range.
It was his seventh goal of the campaign.
Howard then held onto a header from the former Sunderland player from another Ireland centre.
Moyes responded by bringing on striker Victor Anichebe in place of Drenthe after 61 minutes.
Hutton had a volley deflected wide after more good play by Ireland and a cross from Albrighton was headed over by Agbonlahor.
But Anichebe had been on the pitch only eight minutes when he brought Everton level.
Donovan supplied the telling pass and Anichebe kept his cool to slip the ball past Given.
McLeish introduced loan signing Robbie Keane with 10 minutes remaining in place of Warnock with Clark reverting to left-back.
Petrov had a shot deflected wide and Bent had a header punched away by Howard.