Costa Rica, who famously also beat Scotland 1-0 in the 1990 World Cup finals, almost doubled their advantage shortly after halftime but Bryan Ruiz's fierce drive hit the crossbar.
It marked a disappointing return for McLeish, who enjoyed a successful first spell as Scotland manager in 2007.
His side now travel to Budapest to take on Hungary on Tuesday, while Costa Rica meet Tunisia in Nice, France.
(Reporting by Christian Radnedge; Editing by Ian Chadband)
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