Gillespie finished with figures of six for 37 as Durham were bowled out for 248 in their second innings against Yorkshire at The Riverside.
Durham captain Dale Benkenstein (96) had given the home side hope of saving the game but, when he fell to Deon Kruis, the home side collapsed from 205 for five and handed the visitors a 145-run victory.
Sussex missed out on the chance to open up a 15-point lead over Lancashire at the top of division one after failing to force a win against Kent at Hove.
Matthew Walker defied the home side for more than four hours with his 70 not out adding to a first-innings 87.
Sussex's lead is just five points thanks to the efforts of left-hander Walker, who faced 220 balls in his side's innings of 280 for seven, and Tyron Henderson, whose rearguard 31no ensured a draw.
The visitors had been set a total of 457 to win.
In the second division, Essex secured a seven-wicket win over promotion rivals Gloucestershire at Bristol with Andy Bichel, Ravi Bopara and Grant Flower claiming three wickets each.
Former Australia star Bichel took three for 50, medium-pacer Bopara took three for 16 and left-arm spinner Flower returned three for 28 as Gloucestershire were dismissed for 251 shortly after tea.
Division leaders Surrey have a win over Northamptonshire in their sights with the home side 356 runs behind with just four wickets in hand.
Mark Ramprakash (155) and Jon Batty (133) both hit centuries to enable the visitors to declare on 424 for six, leaving Northants a target of 453.
Two wickets each for Azhar Mahmood and Mohammad Akram left Northants in trouble at 97 for six.
The home side had conceded a 28-run first-innings lead and had seen the visitors go from 75 for one overnight to 353 for five as Ramprakash and Batty took control.