Powell, former boss of rugby league giants Leeds Rhinos, has been promoted from his role as offensive coach with the Tykes and he will have responsibility for several areas, including selection and back-play.
Rugby director Phil Davies will continue in his post, with a specific focus on coaching the forwards and strategic development at Headingley.
Tykes announced earlier this season they would be seeking a new head coach.
"Over the past three months, we have done an honest and thorough evaluation of our club and what we need to do to take us forward," said Davies.
"Originally, we spoke about a senior player group, but it became apparent almost immediately that the players were better served concentrating on playing than coaching, and we changed that system after the first week.
"Since then, Daryl has shown a tremendous thirst for learning and has really developed and grown into the role. We feel the time is right for his promotion.
"At the start of the season, Daryl had only recently been introduced to rugby union and, despite having coached at the highest level in rugby league, we felt it would take longer for him to adapt to the game.
"However, over the past eight-to-12 weeks it has become obvious that the right man to take us forward is Daryl. A big part of our decision has been the feedback we have received from the players, particularly the back-line," he added.
"Personally, it has been a very challenging time for me. Having been at Leeds for 10 years, it was only right to take a step back and evaluate what was best for this club.
"I am now more committed than ever to take this club forward to the next level. If anything, the past three months have made me realise my passion and desire to succeed at Headingley is stronger than ever, and hopefully over the coming weeks and months that is exactly what we will do."
Leeds currently prop up the Premiership, two points behind Bath, but they could still reach this season's European Cup quarter-finals if they win their remaining pool games against Calvisano and Cardiff Blues.