The Seagulls took the lead after only 57 seconds as Tommer Hemed's initial shot came back off the underside of the crossbar, but Bobby Zamora was on-hand to tap in from close range.
However, Burnley were awarded a penalty moments later after Lewis Dunk was adjudged to have pulled Michael Keane's shirt.
While the decision initially appeared to be harsh, with both players jostling with each other for space in the area, there was no doubt in Gray's mind as he strode up to dispatch his shot into the right corner.
Brighton pushed forward as they tried to retake the lead and Bruno managed to slide a pass into the path of Solomon March. However, his effort took a deflection and was comfortably gathered by Tom Heaton.