Beltran drove in all of the Cardinals' runs to begin the third consecutive appearance by St. Louis in the league's championship series.
St. Louis survived persistent threats in extra innings by the Dodgers, who will try to even the best-of-seven series when the teams make a short turnaround for Saturday's scheduled start at 4:07 p.m. EDT.
Clayton Kershaw will take the mound for the Dodgers in Game 2 while St. Louis will counter with Michael Wacha.
Los Angeles scored the first two runs of the NLCS on a single to center by Juan Uribe in the third, but the Cardinals came back to tie it in the bottom half of the inning.
After two were out in the St. Louis third, Joe Kelly singled and Matt Carpenter drew a walk off Zack Greinke. Beltran then doubled off the wall in center to bring home both runners.
It took 10 more innings for either team to score, although Los Angeles' inability to produce runs was not for a lack of opportunity.
Los Angeles came up with a one-out triple in the 10th, put runners on first and second with two out in the 11th and had runners on first and second with one out in the 12th. St. Louis escaped each time and finally made the Dodgers pay in the 13th.
With Chris Withrow on the mound for his second inning of work, Daniel Descalso reached on a one-out single. Matt Carpenter then drew a walk and Los Angeles closer Kenley Jansen, who recorded 28 saves during the regular season, came out of the bull pen to face Beltran.
Beltran took a strike and then looked at three straight pitches out of the zone before turning on a fast ball and sending a liner down the right field line. The hit was ruled a single as Descalso trotted home from second with the winning run.
The victory went to Lance Lynn, who dodged trouble in the 12th and then retired the side in order in the 13th.
Thirteen pitchers saw action in the game. The Cardinals managed only seven hits in the 13 innings, two less than the total produced by the Dodgers.
St. Louis lost last year's NLCS in seven games to eventual World Series winner San Francisco. The Cardinals captured the league title in 2011 against Milwaukee before going on to a World Series victory over Texas.
The only Los Angeles appearances in the NLCS in the last 24 years came in 2008 and 2009. The Dodgers lost both times.