Budescu was born in Manasia, a commune near Urziceni in Ialomița County. At the age of 12, with his first team Rapid Sălciile, he once scored "13 goals in one match".
In 2007, Budescu made his debut in Liga II for Petrolul Ploiești whilst still a junior player. Gradually, he imposed himself as a regular first-team member, eventually catching the eye of Rapid București, Politehnica Timișoara and FC Brașov.
In 2009, following the transfer of Laurențiu Marinescu to Unirea Urziceni, Budescu took over the number 10 jersey.
In the 2011 summer, Budescu signed a contract with neighbouring club Astra Ploiești, later relocated and renamed Astra Giurgiu. On 21 July 2013, he scored his first hat-trick in Liga I, in the 4–0 away win against Viitorul Constanța.
On 25 July 2013, during Astra's first ever appearance in the UEFA Europa League, Budescu netted twice in a 2–1 success at Omonia Nicosia as his team played 30 minutes with one player less and qualified to the third round. Due to his exploits, he was dubbed the "Hero of Nicosia".
On 21 August 2014, Budescu scored a penalty against French side Lyon in the first leg of the Europa League play-off (2–1 away win, qualification on the away goals rule). On 6 August of the following year, in the same competition but in the third qualifying round, his second leg brace helped defeat West Ham United 2–1 at home and 4–3 on aggregate.
In January 2016, Astra rejected an offer from Steaua București for Budescu. The following month he signed a three-year contract with China League One club Dalian Yifang for an undisclosed fee, worth €3 million plus 500,000 upon signature.
Budescu was loaned to his previous club in August 2016, being released by the Asians the following January.
On 26 January 2017, Budescu chose to stay once again with Astra Giurgiu, joining on a one-and-a-half-year contract. On 16 February, he netted a goal against Belgian club Genk in a 2–2 Europa League round of 32 home draw.
On 10 June 2017, Budescu signed a four-year deal with Steaua București. He made his debut and also scored in a 2–1 home victory against FC Voluntari on 16 July. Budescu netted again on 25 July, converting a free kick in a Champions League third qualifying round home draw with Viktoria Plzeň.
Budescu played an unofficial match for Romania B in a 1–4 friendly defeat against Poland, in February 2013. His debut for the full side was made two years later, as he came on as a 78th-minute substitute in a 0–0 away draw against Hungary for the UEFA Euro 2016 qualifiers.
On 11 October 2015, Budescu scored a brace against Faroe Islands, helping the country achieve qualification by finishing second in the group one point behind Northern Ireland.
Budescu is regarded as a creative player, with good free kick skills as well as an accurate long shot. However, he is also criticized for his lack of work rate.
Budescu is nicknamed "Sărmăluță", due to his weakness for traditional food. In October 2014, he married Mădălina in Ploiești, with former footballer Dănuț Coman acting as best man.As of match played 14 September 2017.
As of match played 17 November 2015.
Scores and results list Romania's goal tally first.
Cupa României: 2013–14
Supercupa României: 2014
Romanian Footballer of the Year: Runner-up 2015