So instead of describing what your character does and then rolling. First roll and use the result to steer your rp.

This allows the narrative to flow freely. As there are many occasions if you describe first then roll. Where players role play their PC's being this very (insert_attribute/skill_here) proficient person. Then they roll and they don't beat the dc. At this point the DM says despite all that smooth-talking, show of strength, brilliant insight, or whatever have you "Your PC fails for x whatever reason."

With the proposed method players roll first. Player A says I attempt to persuade NPC X. GM says roll for it and the player rolls a:

Success!: Now the player describes how their PC succeeded. "I walk up to NPC x and say "Look you may not like us and that's fair you don't know us. But If you don't let us do X then Y will happen." Or however you want to rp it.

Fail: As long as the DM accepts the description a player can gives endless reasons why their PC fails. "I walk up to NPC x and say "Look you may not like us and thats fair you don't know us. But you're ignoring me aren't you?. . .".

Of course how much narrative power given to the players is determined by the DM. If a DM wants a full on crazy bonkers comedy campaign. Then failures can be describes as: "I walk up to NPC x and say Look you may not like us and that's fair-" As i'm saying this I accidentally shoot fire out of my mouth and burn the NPC's eyebrows off."

Edit: Full on crazy bonkers comedy campaign works with successes as well.

Edit Edit: Thanks for the silver

Have fun out there ^_^