Gus Poyet's side came into the game having won just three of their last 17 games, but went ahead when Dean Hammond, restored to the first XI, nodded in a 15th-minute opener.
Craig Mackail-Smith's 11th goal of the season doubled the lead, before a stunning long-range effort from Wayne Bridge secured all three points.
It was Hammond who broke the deadlock, as the Southampton loanee headed home Andrea Orlandi's inswinging near-post corner to put the away side ahead.
Polish goalkeeper Tomasz Kuszczak, making his 100th English league appearance, then needed to be alert to maintain the advantage as he denied Daryl Murphy's goalbound effort with an outstretched save.
Barnes' flick on presented Mackail-Smith with the opportunity to double the Seagulls' lead after 34 minutes.
The Albion top scorer made no mistake, edging past Aaron Creswell before sending an arrowed effort in off the inside of the post.
The hosts were unbeaten at Portman Road under Mick McCarthy, but struggled to build any momentum against Poyet's men.
Their frustrations were exemplified by Lee Martin's ambitious effort from distance moments before half-time which flew high and wide.
McCarthy's side pressed for a route back into the game after the interval, but found clear-cut opportunities hard to come by.
Guirane N'Daw came closest to giving the home side hope, but his speculative effort failed to hit the target after 53 minutes.
Poyet's men wrapped up the victory when Gary Dicker's pass picked out Bridge who unleashed a superb strike from distance which rocketed past a helpless Stephen Henderson.