The midfielder was introduced after the half-time break in the Robins' 3-1 defeat away at Brentford.
By the time Elliott stepped onto the pitch for his new club
, the hosts were already leading 3-1 at Griffin Park.
By moving to Bristol City, Elliott has linked-up with his former Burnley boss Steve Cotterill and Cotterill has welcomed his arrival, telling his club's official website: "Wade has always been a very fit boy and adding him to our squad in the short-term can only help us.
"I’ve known him since he was at Bashley and then he went to Bournemouth. I took him to Burnley and he was absolutely fabulous for us.
"He is a very intelligent boy and he has already fitted in with the other lads as though he has been here all along. He is a fantastic player and we are looking forward to having him around.
"We've got him for a month but there's scope to extend the loan if it's right for everyone concerned. He offers us good experience."