The reigning Doncaster Rovers Sunday League champions felt aggrieved after conceding a controversial opening goal and then seeing Willie McGhie receive what they thought was a harsh red card before half time.
East London side Baiteze broke the deadlock with Scawthorpe arguing in vain that goalkeeper Danny Porteous had been pushed causing him to drop the ball over the line.
Josh Gibbons equalised from the penalty spot after a foul on Alex Rennie but the visitors went back ahead almost immediately before McGhie got his marching orders for a challenge with his arm that the referee deemed to be reckless.
Baiteze made it 3-1 following the restart but Sam Corner’s lob set up a grandstand finish.
Scawthorpe pushed for an equaliser but were hit on the break by Baiteze whose fourth goal sealed their passage into the semi-finals of the national knockout competition.
Boss James Kaye said: “I’m so proud of the lads and the run we’ve been on. They’ve done the area proud to get to the last eight of a national competition.
"We’ve still got plenty to play for with the Sheffield Senior Cup semi-final on March 20 and the Montagu Cup final on Easter Monday.”