Pollard contributed 29 points, including a try, five penalties, a drop and three conversions, to go with the try of wing Bjorn Basson in the first half and a 70th minute try by centre Jan Serfontein.
It was the Bulls second straight win, while the Cheetahs suffered their first loss after two victories over the Sharks and the Blues.
Holding a 23-6 lead at halftime and scoring the only two tries of the opening half the Bulls appeared in full control, before the Cheetahs hit back with their first try by wing Raymond Rhule to deficit matters to 23-13 two minutes after the break .
Pollard kicked another penalty to extend the Bulls lead to 26-13 before the Cheetahs scored their second try through a rolling maul by eighthman Boom Prinsloo to narrow the gap to 26-20, before the brilliant Pollard once again made it a 12-point game at 32-20 with a penalty in the 61st minute and in the 64th minute with a well-timed drop goal.
The Cheetahs were earlier first on the board in the third minute through a penalty by flyhalf Joe Pietersen after Bulls flank Deon Stegmann was beaten by the ruck for a 3-0 lead.
The Bulls then picked up the pace to level scores through a Pollard penalty after Cheetahs prop Coenie Oosthuizen was blown up at scrum-time for losing his bind, to level matters 3-3 in the 10th minute.
Pollard ensured the early advantage go the way of the men in blue when he sniped through in the 17th minute for the opening try and converted himself to send the Bulls to a 10-3 lead.
Pollard and Pietersen traded penalties (13-6) before a dubious try from a suspect forward pass put Bulls wing Bjorn Basson away for the Bulls second try in the 31st minute for a demanding 20-6 advantage.
With obstruction called by the Cheetahs but ignored by referee Berry Pollard got the chance to kick another penalty on the stroke of halftime for a commanding 23-6 lead.