Sundowns opened the scoring in the 30th minute, as Billiat teed up his Zimbabwean compatriot Cuthbert Malajila from close range with a cross from the right.
However, Thamsanqa Sangweni equalised on the hour with a superb curling free-kick against his former side before getting sent off late on at Nelson Mandela Bay Stadium.
The result leaves Downs in fourth place on the Absa Premiership log with 15 points from nine matches - a point ahead of the mid-table Chilli Boys, and 10 adrift of leaders Kaizer Chiefs, whose PSL record of 15 straight wins was halted by Moroka Swallows in a simultaneous fixture.
Billiat said: "It's really disappointing, the game-plan was to come here for three points, we didn't want to open the gap.
"We fought very hard, unfortunately we couldn't take our chances. I hope we will bounce back soon."
When pressed on their proflicacy up front, Billiat said: "We are working hard on it. We never give up, we will keep pushing and do what we do. But they were very good on their defence.
"Unfortunately we will have to take a point and move forward."