New signing Lee Cook scored in the 59th minute and Orient were good value for their unexpected victory.
Paolo Di Canio's team lost at Preston last Sunday and were then beaten by local rivals Oxford in the Johnstone's Paint Trophy in midweek. Cook's goal brought an end to a 28-game unbeaten home run in all competitions stretching back to August 30 last year.
Orient had two great chances to go into half-time in front but Michael Symes failed to turn a Cook cross on target with only goalkeeper Wes Foderingham to beat after 32 minutes. Kevin Lisbie missed their other opening six minutes later when he missed from six yards after Gary Sawyer had picked him out.
Cook's goal was no more than Orient, who had lost their previous three matches, deserved. Leon McSweeney crossed from the right and former QPR winger Cook was free at the far post to convert.
It ended Swindon's club record run of 12 home matches without conceding a goal - a run dating back to February 21.