Coach Renee Slegers made eight changes from Sunday’s draw with Manchester United, but the English team struggled to find a clinical finish.
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Chloe Kelly had an early goal ruled out after VAR found Stina Blackstenius offside in the build-up.
Despite creating several chances, Arsenal remained frustrated until Selina Cerci was brought down in the box late on.
Caldentey stepped up in the third minute of stoppage time to fire the decisive penalty past Alberte Vingum and secure all three points.
The victory gives Arsenal a winning start to their European campaign, despite their lack of cutting edge for much of the contest.