The victory launched the Pretoria side into fourth place on the Currie Cup log, while the Pumas slipped further down the table to sixth place.
Wet conditions affected the overall flow of the match with a plethora of handling errors characterising the first half.
The Bulls managed to convert the pressure into points as they scored two converted tries, while flyhalf Jacques-Louis Potgieter’s placekicking kept the scoreboard ticking over.
It was, however, Bulls inside centre Burger Odendaal that had the biggest influence on the match as he had a major hand in one of the tries and asked serious questions of the Pumas’ defence.
The Pumas took an early lead through a Justin van Staden penalty conversion in the 13th minute before Potgieter equalise.
Van Staden succeeded with another penalty goal shortly after the Bulls saw a promising attack inside the Pumas 22 fall on the wayside thanks to a knock-on metres from the line.
However, the Bulls were soon rewarded for the pressure on the gainline as towering flanker Jacques du Plessis burst through for a try after bouncing a tackle, with Potgieter converting for a 10-6 advantage.
Two minutes after the restart Potgieter landed a penalty conversion to give his side a seven-point cushion.
Buoyed by the lead, the Bulls played with more freedom and six minutes later industrious prop Dean Greyling scored his side’s second try of the evening.
Greyling’s try was the product of an enterprising run by Odendaal, who stepped his way past the Pumas defenders to take the gap and get his side close to the tryline.
Chasing the bonus point, the Bulls added their third five-pointer when Piet van Zyl picked and dove for the line after replacement hooker Bongi Mbonambi was stopped five metres short, with Potgieter converting.
The Pumas showed some positive signs of scoring their first try of the evening. Rosco Specman appeared to have scored their first try but a review from the television match official showed a knock on earlier to deny the hosts.
The Bulls, however, would not be denied with a minute remaining on the clock captain Deon Stegmann scored from the back of a rampant lineout drive for the bonus point.
Scorers: Pumas 6 (6): Penalties: Justin van Staden Blue Bulls 37 (10): Tries: Dean Greyling, Deon Stegmann, Jacques du Plessis, Piet van Zyl.
Conversions: Jacques-Louis Potgieter (3), Tian Schoeman. Penalties: Potgieter (3).