Anscombe scored all of his side's 29 points with two tries, both of which he converted, as well as five successful penalty kicks as the Blues bounced back from losses to the Crusaders and the Chiefs to claim their opening Super 15 win of the season.
The Bulls outscored the Blues three tries to two, Pierre Spies and Bjorn Basson (two) the men to cross, but it was not enough for the 2009 and 2010 champions who had started their campaign with wins over the Sharks and the Cheetahs.
The Blues began the game in the bottom two of the overall standings but they could hardly have had a better start as 20-year-old first five-eighths Anscombe touched down in the first minute for the opening try of the match.
He duly converted that effort and then added a penalty in the ninth minute to make it 10-0.
The Blues then lost Ali Williams to a yellow card, however, and the Bulls took advantage with a Spies try in the 15th minute. Although Morne Steyn missed the extra points, the South Africa fly-half further reduced the deficit with a 27th-minute penalty to make the score 10-8.
The Bulls could not make any further inroads before half-time, though, and they found themselves trailing 20-8 at the break as Anscombe added a further 10 points to his personal tally in a four-minute spell through a penalty and then a second converted try.
Anscombe continued to dominate the scoreboard and, after Steyn had slotted over a three-point effort at the start of the second half for the hosts, he added two more penalties to leave the Auckland side 26-11 in front heading into the final eight minutes.
The Bulls gave themselves a glimmer of hope when Basson scored their second try, but Steyn again missed with the conversion attempt and Anscombe's fifth penalty of the game in the 77th minute ended any slim hopes of a comeback from the hosts.
The South African side did manage to add another try in the last minute, Basson's score being converted by Steyn on that occasion, and that at least helped the hosts salvage a losing bonus point.
Bulls 23 (Pierre Spies, Bjorn Basson 2 tries; Morne Steyn con, 2 pen), Blues 29 (Gareth Anscombe con, 2 pen) HT: 8-20.