Osvaldo, who arrived at the south coast club in a £15 million ($24 million) move from Italian side Roma, gave the Saints the lead shortly after half-time and two minutes later England striker Rickie Lambert added a second goal.
Victory took Southampton up into fourth place in the table, with promoted Palace left second bottom having collected just three points from their opening six league matches.
Lambert twice went close in the first-half, shooting over the top from 10 yards before his free-kick was turned away by Palace goalkeeper Julian Speroni.
Palace thought they had a penalty approaching the half-hour mark but, to the visible despair of the London club’s manager, Ian Holloway, striker Marouane Chamakh, was booked for diving as he went down under a challenge from Saints’ Poland international goalkeeper Artur Boruc.
The visitors’ woes intensified when the 27-year-old Osvaldo, an Argentina-born Italy international, made it 1-0 to the Saints in the 47th minute.
Southampton attacking midfielder Adam Lallana worked the ball to Osvaldo, who held off two defenders and fire in a left-footed shot for the Saints’ first goal in open play this season.
And two minutes later the 31-year-old Lambert, who once worked in a beetroot-packing factory to makje ends meet, scored the 200th league goal of his career with a stunning 25-yard free-kick that went in off the angle of post and crossbar.
Southampton came close to a third goal when Luke Shaw’s cross hit the far post and was turned in by Jay Rodriguez, but from an offside position.