Puig has won one singles title on the WTA tour and six singles titles on the ITF circuit in her career. On May 26, 2014, she reached her best singles ranking of world number 41. On July 23, 2012, she peaked at world number 263 in the doubles rankings. In May 2014, Puig won her maiden WTA tour title at the 2014 Internationaux de Strasbourg.
In 2007, she entered her first tournament at the Prince Cup, but withdrew from the qualifying draw.
In 2008, she started the season with a surprise run to the final of the Costa Rica Bowl (Grade 3) as a qualifier losing in straight sets to Eugenie Bouchard. She continued her good form, winning two out of three tournaments. Puig continued good results in small Grade 3 and 4 tournaments. In her first Grade 1 tournament in Lexington, she defeated Charlotte Calhoun before going out to Lauren Embree. She continued to participate in bigger tournaments, though not with as much success as she had in the smaller ones.
Puig had her breakthrough season in 2009; she started doing much better in the bigger tournaments. She reached her first Grade 1 final in Casablanca, losing to Mai Grage of Denmark. She continued playing consistently, highlights of the latter months of the year included a semifinal appearance at a Grade B1 tournament in Tulsa, losing to Bouchard in three sets. In her next tournament, she won the doubles title.
In 2010, she continued her successes at a higher level. Puig started the year reaching the final of the 32nd International Casablanca Junior Cup (Grade A), falling easily to Sachie Ishizu of Japan. She then reached another final a week later at the Coffee Bowl (Grade 1) losing to An-Sophie Mestach of Belgium in straights. Puig followed that up with a semifinal appearance at a Grade 1 tournament in Barranquilla. Consistent results followed, before a huge Grade A title at the 27th Copa Gerdau, defeating Jessica Pegula in the final. After a semifinal appearance at another Grade A calibre tournament, she fell in the quarterfinals of the French Open. After that, she failed to advance past the quarterfinals of any of her tournaments until the 2010 US Open – Girls' Singles, where she lost to Yulia Putintseva. Puig finished the season with a flourish, falling in the quarters of Osaka (Grade A), the final of Mérida (Grade 1) and the semis of Key Biscayne (Grade A), both to Lauren Davis.
In January 2012, Puig entered the WTA Tour's ASB Classic in Auckland, New Zealand, ranked 213 in the world. She won the first and second round of qualifying matches, but lost in the last round to the number 145 in the world, Karolína Plíšková. On January 16, Puig entered the 2012 Australian Open qualifying draw ranked 209. She lost in the second round to Kirsten Flipkens. On January 23, Puig entered the $25,000 Andrézieux-Bouthéon ITF tournament as a qualifier. She won two qualifying matches and was able to play in the main draw, but lost in the first round to Magda Linette of Poland.
In the following month, Puig played on the qualifying draw of two WTA Tour events, Bogotá and Monterrey, without being able to get past the first round of the main draw. She received a wild card to play in the 2012 Indian Wells Masters qualifying draw, but, in the second round, lost to world number 66 Eleni Daniilidou. Puig's first 2012 breakthrough came when she reached the final of the Poza Rica $25,000 ITF tournament unseeded. The last two matches prior to the final included wins against tournament seeds numbers four and one, although she lost to second seed Yaroslava Shvedova.
The second breakthrough of 2012 occurred when Puig won the first ITF $50,000 tournament of her career at Joué-lès-Tours, France. She opened the first round by eliminating the number one seed and world ranked 87, Alexandra Panova. In the semifinals, Puig defeated An-Sophie Mestach, the same opponent to whom she lost in the 2011 Australian Open junior final.