The visitors took a 2-0 lead into the interval, with goals from Ramiro Funes Mori and Romelu Lukaku.
But AFC Bournemouth fought back after the break through Adam Smith and Junior Stanislas to seemingly rescue a point, until 21-year-old Barkley slotted late to set the visitors ahead.
Not to be outdone, the Cherries went straight down the other end and rescued a share of the points as Stanislas headed home in the dying moments.
The Cherries came close first after just three minutes. A probing attack resulted in Stanislas crossing for Dan Gosling to try his luck, but the midfielder pulled his shot just wide of the post.
Eddie Howe's men continued on the front foot, and Matt Ritchie soon tried his luck from range, only to be denied by Tim Howard in the Everton goal.
Howard had to be on his toes again shortly after, as King darted through the visitors' defence but had his strike parried and then cleared by the American shot stopper.
Having played on after suffering a potential injury, Cherries ‘keeper Adam Federici then made a fine double to save to deny a header from Everton’s Romelu Lukaku.
But the visitors took the lead after 25 minutes. Ramiro Funes Mori rose above Steve Cook to head an out swinging corner past Federici to put Everton ahead.
Just ten minutes later the visitors doubled proceedings. Gerard Deulofeu played an inch-perfect pass for Lukaku to break onto, and the Belgian took one touch before firing past Federici.
Despite the 2-0 scoreline, the Cherries almost responded immediately through Cook. The defender headed a Ritchie cross goalwards just before the break, but John Stones was on hand to hook the ball clear from the Everton line.
Eddie Howe made two changes at the break. Ryan Allsop replaced the injured Federici and Cook made way for Smith, with Simon Francis slotting in at centre back.
AFC Bournemouth started brightly and King showed an early injection of pace to work space and shoot from the edge of the penalty area, but Howard denied the Norwegian international.
Harry Arter then teed up the late-arriving Charlie Daniels, but his strike was also palmed away by the Everton 'keeper.
Arter then made way for summer signing Glenn Murray on the 75th minute.
With ten minutes remaining AFC Bournemouth found a way back into the encounter. A half cleared corner fell to Smith on the edge of the penalty area, and the substitute rocketed a strike in off the post to reduce the deficit.
Smith really had the wind in his sails and immediately after burst into the penalty area once more. A tangle of legs saw the defender go down in the box, but referee Kevin Friend waved away any calls for a spot kick.
With the momentum of the crowd behind them the Cherries continued to press. King recieved a quick free kick from Murray and bore down the right flank. His cross then found the feet of Stanislas in the penalty area who tucked an equaliser past Howard.
AFC Bournemouth piled on the pressure in the dying moments with a number of dangerous balls into the box, but Everton regained the lead through Barkley with seconds to play.
Not to be outdone, the Cherries broke straight down the other end and Stanislas was on hand to head in a Charlie Daniels header to rescue a share of the points.
Cherries: Federici (Allsop, 45), Francis, Cook (Smith, 45), Gosling, Surman, Arter, Daniels, King, Distin, Ritchie
Subs: MacDonald, Kermorgant, Murray, O'Kane, Cargill
Attendance: 11,228 (1,388)