Zhao Xintong and Noppon Saengkham won deciding frames against the former world champions.
Three-time UK champion Higgins led 5-3 before the 24-year-old Zhao reeled off three frames in a row, clinching his 6-5 win with a nerveless break of 83.
Later Bingham battled back from 5-0 down against Saengkham to force a decider, only to sit and watch as the Thai responded with a cool century to book his place in the last 16.
Higgins told Eurosport: “He was by far the better player, I was pretty poor all day.
“I missed a terrible red when I was 20 in front at 5-4, and when I missed that I knew I was going to lose 6-5. If you don’t take your chances you don’t deserve anything.”
World number one Mark Selby, defending champion Neil Robertson and this year’s world finalist Shaun Murphy have already been dumped out in the early stages of the tournament.
But there was little danger of record seven-time champion Ronnie O’Sullivan joining them as he hit a century and five more 50-plus breaks in a 6-3 win over Essex veteran Mark King.
Kyren Wilson shrugged off Chinese teenager Wu Yize 6-3, while Jack Lisowski was a 6-2 winner over Graeme Dott, who faces a fine after prematurely conceding the sixth frame when he trailed by 24 points with 27 still on the table.