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The hosts allowed the Chiefs to enforce the draw in the final 10 minutes of the match despite a 16-point advantage.
"Anything else than a victory at home feels like a loss, we'll take the two points, it is against the champions but we should have had five," Matfield said.
"With 25 minutes to go at altitude we should have done better."
The Bulls were listless in the first half compared to a spirited Chiefs side that pounced on turn-over ball that came their way through mistakes by the home side.
The tourists cruised to a 15-14 lead going into the half-time break and were unlucky not to have enjoyed a larger advantage at this stage of the match.
The Bulls, however, roused from their first-half slumber to dominate the Chiefs in the half-hour after the break. The side scored three tries --which included a penalty try -- to secure a 16-point lead.
Two tries in the last 10 minutes saw the Chiefs secure three points from the match for scoring five tries, while the Bulls were left with only two.
The Chiefs are in third place on the overall Super Rugby log with 17 points, while the Bulls are in seventh place only a point adrift of the New Zealand side.
"It was there for the taking, I thought we played well the 30 minutes after half-time where we were on top of the Chiefs, we scored some lovely tries and built an innings," said Bulls coach Frans Ludeke.
"But credit to them for the way they came back at us, to score those two tries, they worked hard for it."
In the build-up to the match, the Bulls cautioned against giving the Chiefs turn-over ball, but it proved easier said than done in the clash.
The visitors' opening two tries came off turn-over ball and Matfield admitted his side gave the Chiefs too much possession.
"We actually gave them too much ball to play from, I think if we held on to the ball a bit more, we could have put them under pressure more we should have got that fourth try," he said of the final 10 minutes of the clash.
"With six minutes to go we went for poles, which if you think about it maybe if we went for the corner and kept them in their half and score a try it would have been game over.
"In that last 15 minutes we weren't accurate enough, on attack we didn't build more than three phases and lost the ball and we made silly mistakes on defence."
Chiefs coach Dave Rennie said he was proud of his charges' fightback towards the end to ensure they take three points out of the encounter.
"I am really proud of the boys, it was a helluva comeback with a lot of character shown," he said.
"We lost the kicking battle really, we kicked poorly and we didn't hang on to enough ball and our lineouts struggled in the first half.
"With 11 minutes to go and down by 16 points we are happy to take three points out of it."
Rennie said while his side were not as accurate as was expected of them they nevertheless held on to enforce the draw.
"There are a lot of teams that would have folded today and got a bit of a spanking in the last 10 minutes," Rennie said.
"It is a group of guys that don't know when to give in...today it was good enough to grab three points."
The Chiefs will next travel to Bloemfontein where they will face the struggling Cheetahs, while the Bulls will leave for the Australasian tour on Tuesday.
The Bulls will be hoping for a better tour than last year when they only managed a victory in their first match against the Blues.
They will face the Hurricanes, Highlanders, Waratahs and Force on their tour.
