City's Bernardo Silva and Mateo Kovacic both misfired in the shootout to put Madrid in the driving seat
Eibar, who went into the game sitting one place above the relegation zone, took a shock lead in the 57th minute through striker Kike Garcia before Barca substitute Ousmane Dembele levelled for 10 minutes later.
Messi had returned to Barcelona from Argentina earlier on Tuesday but watched from the stands as he is still recovering from an ankle problem and, while his side played well without him, they lacked the cutting edge he so often provides.
Braithwaite missed the target from the spot in the first half then had a goal ruled offside by VAR while the Dane also missed a chance to grab the winner from point-blank range when the ball bounced off his chest on the rebound and squirmed wide.