Carlisle boss Greg Abbott said he and his players had let the club down following their 4-0 defeat at his former club Bradford.
It was Bradford's first win since gaining promotion to League One through last season's play-offs with summer signing Mark Yeates quickly impressing the home fans in the 13,641 crowd as he scored the first of three goals in nine minutes midway through the first half.
Bermudian international Nahki Wells, who had a short spell at Carlsile before joining Bradford, added the second five minutes later, James Hanson scored the third in the 29th minute and skipper Gary Jones added the fourth goal in the 66th minute.
Abbott, who spent nine years as a player with Bradford, said: "Too many of our senior players let themselves down and we have all let the club down.
"But it is difficult when Bradford are signing Mark Yeates for thousands and we are signing players for hundreds.
"Success comes with a price and I am afraid good teams come with a price. If you are not paying the price you get found out."
Abbott, whose side beat Blackburn at home in the League Cup in midweek on penalties after losing their opening League match 5-1 at home to Leyton Orient, said: "Bradford were better and tougher than us in most departments.
"I am afraid that all over the field we were second best. They were miles ahead of us.
"They were tougher, more experienced and had more nous about the game."