Mark Cooper's side knew they would take over at the summit with at least a point at Highbury Stadium with Bristol City in action in the third round of the FA Cup.
They looked set to pass up that opportunity when Michael Smith saw an early penalty saved and then Josh Morris struck twice in the space of 10 second-half minutes to put Fleetwood in command, but Swindon rallied to deny the Lancashire side.
Andy Williams pulled a goal back 19 minutes from time to take his tally to 15 for the season in League One and Louis Thompson salvaged a point two minutes from the end when he converted a Ben Gladwin cross.
The point moves Swindon above Steve Cotterill's side, who have a game in hand, on goal difference and ensures they have lost just one of their last nine league matches.
A late strike from substitute Colin Daniel ensured Gillingham suffered a 2-1 defeat in their first game since manager Peter Taylor was sacked.
Former England caretaker boss Taylor was shown the door following increasing discontent from supporters and there was more misery for fans of the Kent club at Vale Park.
Caretaker manager Andy Hessenthaler looked to have claimed a point when Cody McDonald cancelled out Ben Williamson's opener in the first half, but Daniel had the final say to lift Vale up to eighth and leave Gillingham just one place above the relegation zone.
Coventry City ended a run of three games without a win in the other League One game on Saturday, beating Walsall 2-0 at the Bescot Stadium courtesy of goals from Jim O'Brien and Marcus Tudgay, who had earlier missed a penalty for the visitors.
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