Bournemouth were also reduced to 10 men just past the hour when Adam Smith received his marching orders, but Josh King halved the deficit from the penalty spot in the 75th minute before Nathan Ake incredibly levelled the scores four minutes later.
Bournemouth were the better team in the first five minutes of the clash, but it did not take Everton too long to settle, with Gylfi Sigurdsson causing problems in the number 10 position.
That said, it was the home side that had the first serious chance of the match in the 16th minute - Callum Wilson firing over the Everton crossbar from inside the penalty box following a smart pick-out from Ryan Fraser.
The biggest talking point of the first half came in an off-the-ball incident, however, which saw Richarlison somewhat controversially handed a straight red card in the 41st minute.
Indeed, the Everton forward tangled with Bournemouth defender Smith during a corner, and the incident continued, with Richarlison ultimately flicking his head towards the right-back, which led to the referee sending the Brazilian from the field.
Everton boss Marco Silva resisted the temptation to make any changes at the break, which meant that Walcott and Sigurdsson dropped into the wide positions to form a 4-4-1 formation.
Despite having a player less, the visitors made the breakthrough in the 56th minute when Walcott broke into a dangerous position before finding the back of the net with a powerful finish.
Bournemouth goalkeeper Asmir Begovic will feel that he should have saved it, but the strike from Walcott was true.
The home side were then reduced to 10 men in the 61st minute when Smith, who had been involved in the Richarlison incident, was handed a red card for a last-man challenge on Walcott.
Begovic was then forced to keep out a Leighton Baines free kick moments later as the match continued to produce big moments.
Everton doubled their advantage in the 66th minute when Sigurdsson's excellent cross was headed home by Keane, with Bournemouth switching off from a quick free kick.
The hosts had a super chance to halve the deficit 20 minutes from time when Wilson broke into the Everton box, but Jordan Pickford was on hand to make an excellent save.
Bournemouth did have one back in the 75th minute, however, with King converting from the penalty spot after Wilson had been bundled to the ground inside the Everton box by Baines.
The Cherries then levelled the scores four minutes later when Ake tapped home from close range after Wilson's header had hit the post and dropped into a dangerous position.
There was a lengthy stoppage in added time when Keane picked up an injury following a collision inside his own box, which eventually led to the centre-back being stretchered off the field.
Play eventually resumed once Keane had departed, and there was one final chance for Everton to win it, but Sigurdsson fired over the crossbar from just outside the Bournemouth box as the points were shared.
BOURNEMOUTH: Begovic; Smith, Cook, Ake, Daniels; Brooks (Ibe 65'), Gosling, Surman (Cook 76'), Fraser; King (Mousset 102'), Wilson
EVERTON: Pickford; Coleman, Holgate, Keane (Zouma 96'), Baines; Davies, Gueye; Walcott (Bernard 85'), Sigurdsson, Richarlison; Tosun (Calvert-Lewin 84')