Stoinis is of Greek descent. He was born in Perth, and represented Western Australia at both under-17 and under-19 level. Stoinis played for the Australian under-19 cricket team at the 2008 ICC Under-19 World Cup. The following year, he represented Australia at the Hong Kong Sixes.

After playing several Futures League matches for the state under-23 side, Stoinis made his List A debut for Western Australia in the 2008–09 Ford Ranger Cup. Both his one-day debut and his Sheffield Shield debut (two days later) came against Queensland at the Gabba. Stoinis played one more Sheffield Shield game and two more Ford Ranger Cup matches during the 2008–09 season, and one in each competition during the 2009–10 season, but was not regularly selected.

In Australia, Stoinis has played club cricket for Scarborough in the Western Australian Grade Cricket competition and for Northcote in the Victorian Premier Cricket. He spent part of the 2012 English season playing for the Peterborough Town Cricket Club in the Northampton Premier League, and in one match took a hat-trick. Stoinis also played five Second XI Championship matches for Kent County Cricket Club during his time in England.

In December 2012, Stoinis was selected in the Perth Scorchers' squad for the 2012–13 Big Bash League season, replacing the injured Mitchell Marsh. In the 2013, Stoinis began representing Victoria domestically, before returning to Western Australia for the 2017–18 season.^[15]

He was signed by the Delhi Daredevils ahead of the 2015 edition of the Indian Premier League.^[16] He was then picked up by the Kings XI Punjab for the 2016 season in the auction for INR 5.5 million.^[17] On 13 May 2016 he achieved his career best T20 figures in a game for Kings XI against Mumbai Indians, taking 4/15 from his four overs.^[18][19]

Stoinis made his Twenty20 International debut against England on 31 August 2015.^[20] His One Day International debut came against the same team on 11 September 2015.^[21] On 30 January 2017, in his second ODI against New Zealand, Stoinis took three wickets and scored 146 not out. This was the highest ODI score from seventh in the batting order by an Australian batsman.^[22] Stoinis was awarded