Steve McClaren's team, who finished sixth and outside the top two for the first time since 2008, seemed to have recovered from three straight defeats when Douglas put them ahead after 14 minutes following a fine combination with Nacer Chadli.
However, RKC equalised 11 minutes later when Ricky ten Voorde converted a penalty after Mitchell Schet was fouled by Dwight Tiendalli.
In the other semi-final, NEC Nijmegen were handed a 3-2 first-leg win over neighbours Vitesse Arnhem thanks to a late own goal by Frank van der Struijk after they twice went behind.
Nicky Hofs had put the visitors ahead after 32 minutes before Navarone Foor equalised four minutes from the break.
Wilfried Bony scored a second for Vitesse but Lasse Schone replied for Nijmegen, who had reached the play-offs on the final day of the season.
The second legs are scheduled for Sunday with the winners meeting in a two-legged final on May 17 and 21 to decide who gets a place in next season's Europa League second qualifying round.
Twente are already assured of a place in the first qualifying round of the Europa League via their Fair Play ranking.