He has earned 45 caps for Slovakia since his debut in 2009, and was included in their squad for the 2010 FIFA World Cup.
On 7 January 2013, he won the FIFA Puskás Award for the best goal of 2012, ahead of Neymar and Radamel Falcao. He bases his game on his two footballing idols Ronaldinho and Ronaldo and tries to emulate them on the pitch.
Stoch's father realized his talent at an early age and placed him with his local club's academy Nitra. His hard work paid off as he was drafted into Nitra's first team after scoring a hat-trick in an under-17 match, at only 15 years old. Due to age restrictions in the Slovak First League, he could not play until his 16th birthday. He made his first-team debut at the age of 16 and by this time he was already being watched by European scouts.
Chelsea had scouts in one of the international competitions he played in and offered him a trial around the same time that Nice was about to sign him. He duly impressed and was signed by Chelsea's youth academy in 2006. He was a regular for the under-18s in the 2006–07 season and scored 11 goals. He ended the season as top scorer for the under-18s even though he had always played on the wing. He started the 2007–08 season well and quickly progressed to the reserve team.
Stoch himself stated that he was close to making his first-team debut and hoped to do so during the course of the 2007–08 season although he did not. Stoch finally made his first appearance as a substitute on 30 November 2008 against Arsenal at Stamford Bridge. He replaced Deco after 80 minutes. Chelsea lost the game 1–2. Stoch made his second appearance for the first-team, replacing John Obi Mikel in the 82nd minute, in Chelsea's 2–1 comeback win over Stoke City, playing a major part in the lead-up to Chelsea's second goal. Stoch again came on for Mikel against Watford in the FA Cup fifth round with Chelsea 1–0 down with 20 minutes. The game finished 3–1 to Chelsea
On 16 July 2009 Twente signed Stoch on loan from Chelsea. Stoch made his first Eredivisie appearance against Sparta Rotterdam. He scored both goals in the 2–0 win over Heerenveen on 20 September 2009. Stoch then scored an impressive opening goal from long-distance, in the 4–0 win against Groningen on 25 October 2009. Stoch continued his impressive scoring record for Twente by scoring and getting an assist in the 2–1 win over Roda on 31 October 2009. In De Grolsch Veste league game against NAC Breda, which Twente won 3–1, Stoch scored two goals to propel the team to victory, after Twente fell behind by a goal. On 2 May 2010, he made the winning goal against NAC Breda, Twente won the match 2–0 in an away game which made Twente the champion of the Eredivisie for the first time. Stoch ended the season with 10 league goals (12 in all competitions), having started almost every competitive match for Twente.
Stoch joined Turkish team Fenerbahçe on 11 June 2010 on a four-year deal for a fee of €5.5 million. On 28 July 2010 he scored his debut goal for the club in a Champions League match against Young Boys in Third qualifying round and Fenerbahçe draw the match with score 2–2 at Stade de Suisse on Bern. Fenerbahçe won the Süper Lig title in 2010–11, for a record 18th time, with Stoch contributing two goals. Fenerbahçe gave Stoch a two-year contract extension in 2012. Stoch is contracted to the club until 2015–16, making €1.75 million and €15000 per match in 2012–13. He will earn €1.75 million in addition to €16000 per game in 2013–14, €1.8 million plus €16000 per match in 2014–15 and €1.9 million as well as €17000 per game in 2015–16. In 2013, he won goal of the year 2012 against Gençlerbirliği on Süper Lig in the FIFA Puskás Award defeating Falcao and Neymar.
In August 2013 Stoch joined PAOK after Fenerbahçe and PAOK came to an agreement for a 1-year loan for €1 million. He scored his first goal with the club in the first leg of the UEFA Champions League Play-off qualifying round against FC Schalke 04. The match ended 1–1.
On July 3, 2014, he signed new contract with Fenerbahçe until 2018 and then joined Al Ain on a one-year loan deal for €1 million.
On 22 August 2015 joined Bursaspor after Fenerbahçe and Bursaspor came to an agreement for a 1-year loan. In summer 2016, he returned to Fenerbahçe.
On 11 August 2017, Stoch joined Czech club Slavia Prague. He has refused offers from Spain, France, Russia or wealthy clubs at Near East. The main reason of this choice was Slavia's big chance of qualification into 2017-18 group stage of UEFA Champions League via upcoming play-off round against APOEL. Stoch has played in both play-off matches but did not helped his new club to qualify to group stage. After the 0-2 loss in Cyprus and 0-0 draw at home, in which Stoch has been subbed off after half-time because of injury, Slavia was eliminated from the Champions League and entered 2017–18 UEFA Europa League group stage.
Stoch made his senior début for Slovakia on 10 February 2009 against Ukraine, where he came on in the 70th minute. His second appearance came just 24 hours later against Cyprus. He scored his first international goal for Slovakia against San Marino in a 2010 World Cup qualifier on 6 June 2009. Stoch was selected for the 2010 FIFA World Cup squad. On 7 September 2010, he scored the only goal of a Euro 2012 qualifying victory against Russia in Moscow. On 9 October 2014, Stoch scored the game-winning goal in the 2–1 UEFA Euro 2016 qualification win against title-holders Spain three minutes from time. Stoch became a part of a preliminary EURO 2016 squad and capped in a match against Germany, only days after death of his mother, due to which he had to be excused from a training camp in Austria in the previous week.As of 8 June 2017
FA Cup: 2008–09
Süper Lig: 2010–11
Turkish Cup: 2011–12, 2012–13
UAE Pro League: 2014–15
FIFA Puskás Award: 2012
Peter Dubovský Trophy: 2009, 2010