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. 
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 August 2016, he made his debut for Sevilla in the UEFA Super Cup final. They played Champions League winners Real Madrid in an all-Spanish final. Unfortunately Vietto and Sevilla were beaten 3-2 after extra time. He made his La Liga debut for Sevilla on 20 August. In a perfect start to his La Liga career, he scored 2 goals in the 6-4 win against Espanyol. On 17 September 2016, he scored again, this time in the 1-1 draw with SD Eibar. On 29 October, he scored a goal in the 1-1 draw against Sporting Gijon. On 2 November, he scored in the 4-0 Champions League Group H win against Dinamo Zagreb. On 17 December 2016, he scored 2 goals in the 4-1 win against Malaga. On 21 December, he scored a hat-rick in the amazing 9-1 Copa Del Rey win against Formentera. In total, he played 31 games, scoring 10 goals in his season long loan at Sevilla.
Return to Atletico Madrid
On his return to Atletico, he was mainly used as a substitute, although he did come on in many games. On 9 January 2018, he scored a hatrick in the 4-0 Copa Del Rey win against Las Palmas. On 11 February, he scored his first La Liga goal since returning to Atletico in the 3-1 home win against Levante. On 12 May, he scored his fifth and final goal of the season. It was the penultimate game of the season and he scored the only goal of the game as Atletico beat Girona 1-0. It was his last goal for the Spanish club before they loaned him out again, this time to Fulham for a season.
After signing for Fulham on 9 August 2018, he made his debut just 2 days later. It came in the 2-0 Premier League defeat to London rivals Crystal Palace.
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 
|Racing Club||2011–12||Primera División||2||0||1||0||—||3||0|
|Atletico Madrid||2015–16||La Liga||19||1||4||1||5||1||28||3|
|Sevilla FC||2016–17||La Liga||21||6||5||3||5||1||31||10|
- Atletico Madrid
- In isolation, Darío and Vietto are pronounced [daˈɾi.o] and [ˈbjeto] respectively.