It was the first trip to Busch Stadium for the Rangers since they came within one strike of winning the 2011 World Series. St. Louis eventually captured the title with home wins in the sixth and seventh games.
The teams combined for eight runs and eight hits over the first two innings while battling to a 4-4 tie and that was all the scoring until Texas broke through in the ninth.
Ian Kinsler singled to open the ninth off Trevor Rosenthal (1-1) and Rosenthal then mishandled a bunt by Elvis Andrus to put runners on first and third. A wild pitch moved Andrus to second and both runners scored when Cruz singled to center.
It was Nathan's 23rd save and he protected the win for Neal Cotts (4-1), who worked out of a bases-loaded, two-out jam in the eighth.
St. Louis scored three times in the first, but the Rangers bounced back with four in the second thanks in part to a two-run single by Andrus.
The Cardinals had won their previous two games.
Chipotle plans first price increase in 3 years
Justin Bieber crashes Drake Bell's album release party