Goals from new signing Stephen O'Leary and Bradley Hudson-Odoi in an eight-minute spell early in the second half deservedly settled the game in the Bulls' favour.
An entertaining first half produced chances at both ends but no goals.
Crewe were first to show with a shot from Tom Pope well saved by the returning Darren Randolph.
An excellent tackle by Richard Rose on Joel Grant retrieved a dangerous position, before Hereford carved out their best chance of the half.
Steve Guinan sent Rose away on the right and the full-back's fine cross picked out O'Leary, who headed over from a good position.
Crewe looked unfortunate to be denied a penalty just before the interval with referee Pat Miller spotting Dean Beckwith's foul, but awarding a free-kick on the edge of the area.
Randolph had to pull off a flying save to tip the ball over when Michael O'Connor's shot beat the defensive wall.
After a lively start to the second period, Hereford took the lead in the 48th minute.
Ben Smith's cross was half cleared to O'Leary 25 yards out and the former Luton Town man fired a superb shot into the top corner.
O'Leary turned provider minutes later, delivering a pass which Hudson-Odoi clipped neatly into the Crewe net.
And it was almost three shortly afterwards when only a superb save by Steve Collis prevented Dean Beckwith's header flying in.
Hereford continued to dominate for the remainder of the game and the passing football, which has been their trademark in recent seasons, was much in evidence.
Their win might have been even more convincing had O'Leary's late shot, which flashed past a post, been directed a yard to the right.