It was the visitors who made the better start, but they were punished for not scoring while on top in the 20th minute.
The Clarets came close to equalising when Jason Shackell's header was cleared off the line.
Tomasz Kuszczak, who made a number of stops against Cardiff City in midweek, pulled off another impressive save to deny Shackell once again.
Leonardo Ulloa missed a glorious opportunity to put his side further in front just before half time as he fired Orlandi's cross over the bar from point-blank range.
Burnley continued to put pressure on Brighton's defence, but could not find a route back into the game.