The former Chelsea striker got the 40th-minute opener as the American club he captains and co-owns beat Whitecaps FC II 2-1.
Ex-Newcastle midfielder Tiote died after collapsing in training in China.
"I want to want dedicate this to Cheick Tiote who passed away doing what he loved," Drogba, 39, wrote on Instagram.
"My thoughts and condolences go out to his family, the Ivory Coast and the whole of Africa, mourning the passing of one of their own."
Previously known as Arizona United, Phoenix hope to become one of four planned expansion teams in the MLS over the next three years.
They are in their fourth season in the Western Conference of the United Soccer League, which features several MLS reserve sides.
Drogba, who had not played since leaving MLS club Montreal Impact in November, also set up former England winger Shaun Wright-Phillips, 35, for a 77th-minute winner.
Ivory Coast's record goalscorer hit 157 goals in 341 appearances during his first spell at Chelsea from 2004 to 2012, winning three Premier League titles and the Champions League.
Following moves to Shanghai Shenhua in China and Turkish side Galatasaray, he returned to the Blues for the 2014-15 season, scoring seven goals in 40 appearances, helping Jose Mourinho's side to another league title, before 18 months with Montreal.
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Didier Drogba scored on his Phoenix Rising debut and dedicated the win to his former Ivory Coast team-mate Cheick Tiote, who died on Monday aged 30.