Puig advanced to the final of two junior Grand Slams: 2011 Australian Open (l. Mestach) and 2011 Roland Garros (l. Jabeur). In her fourth career tour‐level main draw appearance (2013 Brisbane), defeated Rogowska in opening match, becoming the first player from Puerto Rico to win a main draw match since 2006 Charleston (Kristina Brandi). In Oeiras she reached her first WTA quarterfinal and at Roland Garros, on Grand Slam debut, she became first woman representing Puerto Rico to reach 3r at a Grand Slam in the Open Era.
In 2014 she played and won her first WTA final in Strasbourg, defeating Sílvia Soler Espinosa. She became the first player representing Puerto Rico to win a WTA singles title; Kristina Brandi and Gigi Fernández, both born in Puerto Rico, won titles while playing under USA flag. She finished the year winning the inaugural Rising Stars Invitational as part of WTA Finals in Singapore (d. Zheng Saisai in F).
After a disappointing 2015 season, she made history at Rio Olympics, winning Puerto Rico’s first ever Gold medal. Puig’s first upset came in the second round, where she beat world No. 19 Anastasia Pavlyuchenkova in straight sets. She offered a dominating performance to beat reigning French Open champ Garbine Muguruza, toppled No. 32 Laura Siegemund and No. 14 Petra Kvitova, and finished the job against No. 2 Angelique Kerber in the gold-medal match. She reached at least quarterfinal on eight occasions during the year, also in Sydney (R‐Up; l. Kuznetsova in first final at Premier level), Prague (QF, l. Kuznetsova), Nottingham (SF, l.Ka.Pliskova), Eastbourne (SF, as qualifier, l. Cibulkova), Florianopolis (SF, l. Begu), Tokyo [PPO] (QF, l. A.Radwanska) and Tianjin (QF, l. Kovinic).
In 2016 she saved saved five match points during a second round win to Krystina Pliskova, who served a WTA‐record 31 aces in the match, and equalled her best Grand Slam performance. She also scored her best result at WTA Premier Mandatory tournament with a third round run at Indian Wells and made her Top 30 debut the week of September 26, 2016.
Puig was unable to replicate her success throughout 2017 and fell out of the top 50 on 12 June. She concluded her 2017 season without advancing past the second round of any Grand Slam or WTA Premier Mandatory tournament, however she reached her fourth career WTA final in Luxembourg.