Vietto was born in the small town of Balnearia in the Province of Córdoba. He joined his local team Independiente de Balnearia at the age of seven.
At the age of 15, Vietto joined Argentine Primera División club Estudiantes, but was released two years later. He subsequently moved to Racing Club's youth academy, after a failed trial at Rosario Central.
In 2011 Vietto signed his first professional contract, and was called up to the main squad by manager Diego Simeone on 25 October. He made his professional debut on a day later, coming on as a late substitute in a 1–1 home draw against Lanús.
Vietto was handed his first start on 3 September 2012, scoring his first professional goals (and his first hat-trick) in a 3–1 home success over San Martín de San Juan. He became a regular starter under Luis Zubeldía and contributed with 13 goals during the campaign, as his side only finished fifth.
After attracting interest European clubs such as Premier League club Liverpool and Serie A champions Juventus, Vietto signed a four-year contract extension with Racing on 26 March 2013.
Vietto appeared in 35 matches in 2013–14, scoring five times. Highlights included a brace in a 3–1 away win against Gimansia La Plata on 2 November 2013.
On 4 August 2014, Vietto signed a five-year deal with La Liga side Villarreal CF, for a rumoured €5.5 million fee. He made his debut for the club on the 21st, coming on as a second-half substitute for Ikechukwu Uche in a 3–0 Europa League play-off victory over Astana, before scoring twice in the return leg at home.
Vietto made his La Liga debut on 24 August, replacing Giovani dos Santos for the last 10 minutes of a 2–0 away win against Valencian rivals Levante UD. On 21 September he scored his first goals in the competition, netting a brace in a 4–2 home win over Rayo Vallecano.
Vietto added another double on 21 December, in a 3–0 win against the Deportivo de La Coruña also at the Estadio El Madrigal. He finished the month with three goals, with his side moving to sixth position, and earning him the honour of La Liga Player of the Month.
On 22 June 2015, Villarreal confirmed Vietto had joined Atlético Madrid for a fee reported to be in the region of €20 million. Vietto scored his first goal for Atlético on 4 October 2015 as he tapped in a cross from Jackson Martínez to equalize against archrivals Real Madrid in a league match that eventually ended 1–1.
On 30 July 2016, Atlético Madrid and Sevilla FC reached an agreement for the loan of Vietto with an option to buy.
On 9 January 2013, Vietto made his international debut for the Argentina national under-20 team against Chile at the year's South American Youth Championship. He scored his first goal four days later, but in a 1–2 loss against Paraguay, and finished the tournament with two goals in four games.As of 2 June 2017
UEFA Champions League: Runner-up 2015–16
La Liga Player of the Month: December 2014
UEFA Europa League Top assists: 2014–15