The away side should have taken the lead inside the opening 10 minutes, but Andrew Crofts squandered a glorious opportunity following good work by Will Buckley on the right wing.
McGoldrick doubled his tally in the 24th minute when he surged past Greer and drilled a low finish into the far corner.
The former Nottingham Forest striker almost secured his hat-trick after the break, but he poked his effort over the bar after being played in by his strike partner Daryl Murphy.
Murphy himself went close on the hour mark with a towering header from Aaron Creswell's corner, but his effort dropped marginally over the crossbar.
Brighton appealed for a penalty when Buckley's strike seemed to hit an Ipswich hand, but the referee waved play on.
LuaLua looked to be the away side's greatest threat and the winger went close with a fierce effort that hit the side-netting at the near post as Brighton left Suffolk without a point.
The result means Ipswich are now the Championship's highest-scoring side at home.