Daňo played in the 2007 Quebec International Pee-Wee Hockey Tournament with a youth team from Bratislava. He was ranked twelfth among International Skaters by the NHL Central Scouting Services in their final rankings for the 2013NHL Entry Draft. He was selected in the first round, 27th overall by the Columbus Blue Jackets. Daňo returned to Slovan Bratislava of the Kontinental Hockey League for the 2013–14 season. On March 14, 2014, the Blue Jackets signed Daňo to a three-year, entry level contract. Daňo made his NHL debut on October 9, 2014, against the Buffalo Sabres and scored his first career NHL goal, assist and point two days later, on October 11, his second game, against the New York Rangers. He played eight games with Columbus to begin the 2014–15 season before being assigned to the Springfield Falcons, the Blue Jackets' American Hockey League affiliate. In February 2015, however, he was recalled back to Columbus, where he played for the rest of regular season. Daňo finished the NHL season with eight goals and 13 assists in 35 games played, as well as the AHL season with 11 goals and eight assists in 39 games. On June 30, 2015, Daňo, along with Jeremy Morin, Artem Anisimov and Corey Tropp, were traded to the Chicago Blackhawks in exchange for Brandon Saad, Alex Broadhurst and Michael Paliotta. After attending the Blackhawks training camp in preparation for the 2015–16 season, Daňo as a key piece in the Saad trade was surprisingly reassigned to AHL affiliate, the Rockford IceHogs to begin the year and continue his development. On November 2, 2015, Daňo was recalled by the Blackhawks to the NHL. He made his Chicago debut that night in a 4-2 victory against the Los Angeles Kings. He featured in 13 games with the Blackhawks scoring a goal and an assist before he was returned to the IceHogs. On February 25, 2016, Daňo and the Blackhawks' first-round pick in the 2016 NHL Entry Draft were traded to the Winnipeg Jets in exchange for Andrew Ladd, Jay Harrison and Matt Fraser. In 21 games for the Jets, he recorded four goals and eight points in 21 games. The following season, Daňo had eleven points in 38 games. He also skated in six games for the team's AHL affiliate, the Manitoba Moose. On June 13, 2017, the Jets re-signed Daňo to a one-year contract worth $850,000. On July 24, 2018, Daňo re-signed with the Jets, to a one year contract, worth $800,000. After beginning the 2018–19 season as a healthy scratch, on October 14, 2018, the Jets placed Daňo on waivers. He was claimed by division rivals, the Colorado Avalanche the following day. Daňo went scoreless in eight games for the club with 7 penalty minutes before on November 22, he was placed back on waivers. The following day, he re-joined the Jets when they claimed him and was immediately re-assigned to the AHL to join the Manitoba Moose. In remaining in the AHL for the remainder of the season, Daňo contributed with 12 goals and 30 points in 51 games for the Moose. On June 25, 2019, Daňo's tenure with the Jets ended after he was not tendered a qualifying offer, releasing him as a free agent. He signed a 1-year, 2-way contract with the Blue Jackets on 8 August.
