Easton settled the match two minutes from time with a well-taken finish after Dave Edwards seemed to have earned Luton a point with his first for the club.
Midfielder Easton arrived unnoticed to fire home at the back post after Lee Peacock had diverted substitute Ibon Arrieta's cross into his path.
Swindon had taken the lead 11 minutes into the second half through Scottish striker Peacock.
Frenchman Sofiane Zaaboub whipped in an inviting cross from the left for Peacock to head into the top corner from four yards.
Luton were level just ten minutes later when Edwards also picked out the top left corner with an even better header.
Former Ipswich winger Darren Currie crossed from the left and Edwards nodded home from 12 yards.
Swindon, who were held to a 1-1 stalemate by Northampton on the opening weekend, were unlucky to have Hasney Aljofree's 27th minute header ruled out after the referee spotted a shove.
Aljofree also felt his 40th minute effort had crossed the line before midfielder Steven Robinson cleared.
Former Tottenham centre-back Chris Perry came to Luton's rescue in the 59th minute with another dramatic goalline clearance.
The impressive Zaaboub picked out Roberts at the back post and his volley was destined for the bottom corner before Perry stepped in.
Luton's veteran strike pairing of Paul Peschisolido and Paul Furlong caused Swindon problems early on.
Furlong was unlucky with a couple of headers that just missed the crossbar.
The victory leaves Swindon with four points from the opening two games, while Luton have one win from their opening encounters.