On June 2, 2013, the Dodgers announced that they had recalled Puig from Chattanooga and he made his debut on June 3. In his first at-bat he singled to left-center off Eric Stults of the San Diego Padres. He was 2 for 4 in the game and showcased his strong arm by recording an outfield assist on a double play to end the game.
Puig hit two home runs, doubled and drove in five runs in his second game. He became the first Dodger ever to have a multi-homer game in one of his first two appearances. The first home run was off Padres pitcher Clayton Richard. In his fourth game, on June 6, 2013, he hit a grand slam off of Cory Gearrin of the Atlanta Braves. The following day, Puig hit another homer to become only the 2nd player in the modern era (Mike Jacobs in 2005 was the other) to hit four home runs in his first five games and his 10 RBI tied the Major League record for most RBI in the first five games of a career (with Danny Espinosa (2010) and Jack Merson (1951)). Puig was named the National League Player of the Week for the week of June 3-9.
Puig had 27 hits in his first 15 games, tied with Joe DiMaggio (1936) and Terry Pendleton (1984) for second most all-time behind Irv Waldron (1901) and Bo Hart (2003) with 28. He also became the first player in major league history to record at least 34 hits and seven home runs in his first 20 games and set Dodger records for most hits through 20 games (one more than Gibby Brack in 1937) and total bases through 20 games (58, four more than Del Bissonette in 1928). Puig finished the month of June with 44 hits, breaking Steve Sax's 1983 team record for most hits by a rookie in one month. That total was also second all-time for rookies in their first month, behind only DiMaggio who had 48. In 26 games in June, Puig hit .436 with a .467 on-base percentage and a .713 slugging percentage. He won both the National League Rookie of the Month Award and the National League Player of the Month Award, the first time someone had won both in their first month in the Majors.
Despite his not making his debut until June, Puig received 842,915 write-in votes from the fans for the 2013 Major League Baseball All-Star Game. His name was included among the All-Star Final Vote candidates. Puig received 15.5 million votes, second to Atlanta Braves' first baseman Freddie Freeman who received 19.7 million.
In 2013, Puig hit .319 in 104 games with 19 home runs and 42 RBI. He was selected by Baseball America to their annual "All-Rookie team" and finished second in the National League Rookie of the Year voting to fellow Cuban Jose Fernandez.