World number 43 Evans took a final-set tie-break against Thiago Monteiro to triumph 6-7 (4-7) 6-2 7-6 (7-2).
The 26-year-old's only two previous wins on the surface at tour level came in Davis Cup dead rubbers.
Edmund, the world number 42, brushed aside Frenchman Jeremy Chardy 6-3 6-4 and faces Austria's Dominic Thiem next.
Evans will play Mischa Zverev, who beat Andy Murray at this year's Australian Open, in the second round.
His most recent Davis Cup win on clay came earlier this month as GB lost to France in the quarter-finals.
He was outplayed by Chardy in his first Davis Cup singles rubber, before beating Julien Benneteau as the match descended into chaos.
Much of the build-up to the tie focused on Evans' inexperience on clay, and dislike for the surface.
British number two Edmund, who also lost to Chardy in the singles at the Davis Cup, exacted revenge on the world number 70 with a straight-set victory in Barcelona.
Britain's Dan Evans claimed his first ATP Tour win on clay to reach the second round of the Barcelona Open, as compatriot Kyle Edmund also progressed.