Nicky Ajose had put the hosts in the lead midway through the first half at the County Ground, but the visitors equalised immediately through David Mooney.
Strikes from Moses Odubajo in the 40th and 46th minutes sealed the points for Russell Slade's side, while Swindon's miserable day was compounded when Jay McEveley saw red for a second caution in the final minute.
That result saw Orient leap above Wolves, who were held 0-0 at home by fourth-placed Brentford, and there was a fourth consecutive defeat for third-placed Peterborough United, who went down 1-0 at home to struggling Stevenage thanks to Lucas Akins' 20th-minute strike.
Nahki Wells put Bradford ahead on 21 minutes before they were pegged back by Patrick Bamford's strike 15 minutes later.
At the wrong end of the table, Notts County remain bottom after Adam Reach's 73rd-minute goal earned Shrewsbury Town a 1-0 win, while Crewe Alexandra in 23rd also fell to defeat, 2-1 at home to local rivals Port Vale.
The hosts took the lead 10 minutes before the break through George Evans but Rob Taylor's strike two minutes before the break and Jennison Myrie-Williams' winner seven minutes from time ensured Crewe's winless run extended to five matches.
The other two sides in the relegation zone, Bristol City and Sheffield United, went head to head at Ashton Gate with Aden Flint's own goal 12 minutes from time proving enough for the visitors to jump above their opponents into 21st place.
Tranmere's Ash Taylor was dismissed two minutes later but it had little impact on the visitors, who rounded off the scoring through James Wallace a minute before the end.
Elsewhere, Michael Petrasso's 57th-minute strike handed Oldham Athletic a 1-0 win at Gillingham, while there were goalless draws in the games between Crawley Town and Walsall and Rotherham United and Carlisle United.
