The South American giants, missing injured captain Lionel Messi, wrapped up their two-match tour of the United States with a deserved victory at the Los Angeles Memorial Coliseum.
Lionel Scaloni's side, who defeated El Salvador 3-0 in Philadelphia on Friday, dominated the opening 45 minutes and carved out a series of promising openings.
However it was Costa Rica who took the lead on 34 minutes after a sweeping counter-attack.
Alvaro Zamora latched on to Manfred Ugalde's pass and fired a low shot that was blocked by Argentina goalkeeper Walter Benitez.
Benitez's save however fell invitingly into the path of Ugalde, who bundled home the rebound for 1-0.
Argentina got back on level terms early in the second half with a superb equaliser from veteran Di Maria on 52 minutes.
The 36-year-old Benfica winger hit a superb free-kick from just outside the area which flew into the top corner beyond the reach of Costa Rica goalkeeper Keylor Navas.
Argentina made it 2-1 four minutes later from another set-piece.
Nicolas Tagliafico headed against the bar after a corner and Liverpool midfielder Mac Allister reacted first to nod home the rebound from close range.
Inter Milan forward Martinez made the game safe in the 77th minute, latching on to a delightful through ball by Rodrigo De Paul before lifting a deft finish over Navas.
