Jordan Pickford stopped Everton sinking to a fourth straight defeat by the narrowest of margins as he turned Oli McBurnie's shot on to the bar in stoppage time to ensure the points were shared.
McBurnie looked as though he would win it for Sheffield United and pile the pressure on Sean Dyche when he headed goalwards in the ninth minute of stoppage time, only for Pickford to fly to his right and keep it out.
That made up for Pickford's unlucky own-goal at the end of the first half.
Though both ended runs of three successive defeats, they remain winless and did little to suggest that they can steer clear of an extended relegation battle this season.