The visitors were always in command after Eduardo’s 25-yard free-kick had deflected off Younes Kaboul (28), before Samir Nasri (42) drilled in a second just before half-time.
Wales teenager Aaron Ramsey smashed home a fine third on 69 minutes when he skipped round two tackles before drilling home left-footed from the edge of the box.
Although Nadir Belhadj – one of several Pompey players set to leave for African Nations Cup duty – pulled a goal back (74) at the end of a well-worked move, Cameroon midfielder Alex Song powerfully headed in a fourth with nine minutes left.
The win moves Arsenal to within four points of leaders Chelsea, with the Gunners boasting a game in hand. Pompey remain bottom.