A former Bulgaria under-21 international, he was signed to Chavdar Etropole from Volov Shumen in 2009. He made his first-team debut during the 2010–11 season, before he joined English club Nottingham Forest in April 2011. From Forest he has spent time on loan at Ilkeston, Gainsborough Trinity, Nuneaton Town, Mansfield Town, Olhanense (Portugal), and Port Vale. He was named as the Championship Apprentice of the Year for the 2012–13 season, however he left the club in August 2018 after making a total of just three first-team appearances. He signed with Burton Albion in September 2018, and then moved on to Accrington Stanley four months later. He returned to Bulgaria to sign with CSKA Sofia in July 2020 and won the Bulgarian Cup at the end of the 2020–21 season.
Evtimov signed to Chavdar Etropole from Volov Shumen in 2009. He made his first-team debut in a 5–1 victory over Kom on 18 September 2010. He made a total of five appearances in the B Group in the 2010–11 season.
Evtimov signed for English club Nottingham Forest in April 2011. He spent a brief time on loan at Ilkeston, before he joined Gainsborough Trinity in the Conference North on a one-month loan on 10 November 2011. He played three matches in the space of a week for Steve Housham‘s “Holy Blues”, before losing his first-team place at The Northolme. He was named as the Championship Apprentice of the Year for the 2012–13 season. On 16 July 2013, he joined Conference Premier club Nuneaton Town on a three-month loan deal. He made four appearances during his stay at Liberty Way. He made his first-team debut for Forest on 21 April 2014, coming on as a late substitute for the injured Dorus de Vries in a 2–0 win at Leeds United. He signed a new two-year contract with the club three months later after new manager Stuart Pearce was told by caretaker-manager Gary Brazil that he was a good prospect.
A loan to Wrexham fell through in July 2014 after manager Kevin Wilkin was unable to accept an instant recall option demanded by Forest due to regulations by the FAW. Evtimov instead joined Nottinghamshire League Two club Mansfield Town on 19 August, on loan until 3 January, after impressing manager Paul Cox during pre-season. He was named on the Football League team of the week for his performance in a 1–0 win at AFC Wimbledon on 20 December. He played a total of 11 games for the “Stags”, and new manager Adam Murray stated that “we attempted to extend his loan but unfortunately for us, Forest see him as part of their long-term plans”. Evtimov extended his contract with Forest by a further four years in July 2015, tying him to the club until 2019. Following an injury to de Vries, he made his second appearance for Forest at Sheffield Wednesday on 31 October 2015, and manager Dougie Freedman said that he “cost us the game” after he allowed a strike from Fernando Forestieri to creep underneath him for the only goal of the game.
On 31 August 2016, he signed on loan with Portuguese club Olhanense for the 2016–17 season, who were bottom of the LigaPro. His loan spell at the Estádio José Arcanjo proved a difficult one as he conceded within three minutes of his debut, in a 3–2 defeat at União da Madeira on 10 September. Manager Cristiano Bacci lost his job the following month, and though Evtimov established himself in the first team under new boss Bruno Baltazar, he was recalled from his loan spell earlier than planned on 13 January, having made ten appearances for the “Lions”. Evtimov started the 2017–18 season as Mark Warburton‘s third-choice goalkeeper behind Jordan Smith and Stephen Henderson, and was advised by former “Reds” goalkeeper Barry Roche to go out on loan to find first-team football. On 8 August 2017, Evtimov started Forest’s first round EFL Cup tie against Shrewsbury Town, conceding a penalty in a 2–1 win. This was Evtimov’s first game for Forest since October 2015, and his home debut at the City Ground six years after first joining the club.
On 28 October 2017, Evtimov joined League Two club Port Vale on an emergency loan following an injury to loanee goalkeeper Kelle Roos, and made his “Valiants” debut later that day against Swindon Town at Vale Park. At fault for at least one of the goals in a 3–0 defeat, The Sentinel‘s Michael Baggaley wrote that “Evtimov looked as though he had only just been introduced to his defence, but that’s not a criticism because actually he had”. Manager Neil Aspin reserved his criticism for the club’s medical staff as they only ruled Roos as unfit to play the previous day. On 31 August 2018, Evtimov’s contract with Nottingham Forest was terminated by mutual consent after he found himself behind Costel Pantilimon, Luke Steele and Jordan Smith in the first-team pecking order.