The home side started brightly, with Jordi Gomez and Nathan Dyer missing presentable chances inside the opening 10 minutes.
With Brighton struggling to get into the game, it was no surprise to see Swansea go ahead after 17 minutes. Andrea Orlandi's free-kick caused confusion in the visitors' area and after Alan Tate's header had cannoned off the post, Monk was on hand to nod home the rebound.
Brighton responded immediately with Adam Virgo heading over from Dean Cox's free kick, before Dobbie fired wide for the Swans from 20 yards.
Mark Gower spurned a glorious chance to double Swansea's lead seven minutes before the break as he latched on to Dyer's cross only to drag his shot wide from close range.
The Swans did get their second goal on the hour mark, Dobbie latching on to Orlandi's cross and firing home at the second attempt after his initial header had been blocked by Andrew Whing.
Substitute Guillem Bauza spurned a number of chances to extend Swansea's advantage in the closing stages, but the Spaniard made amends in injury-time by supplying a perfect cross for Dobbie, who made no mistake from 12 yards.