The Bulls extended their own unbeaten run to four matches thanks to Andy Ferrell's strike five minutes before the interval.
Morecambe were unlucky to be denied a penalty in the sixth minute when Alex Jeannin appeared to bring down Garry Thompson in the area only for referee Snartt to award the free-kick outside.
The visitors dominated the early stages but Hereford gradually clawed their way back and started to create openings.
Craig Stanley drove a long-range shot wide and should have done better when he raced on to a Dean Beckwith pass only to blast wide of an upright.
Stanley's header from a Ferrell cross struck Michael Howard's hand but referee Smartt waved play-on.
The decisive goal arrived on 40 minutes when Jeannin's cross-shot was half-cleared by Chris Blackburn to Ferrell who lashed a left-foot volley into the bottom corner of the net.
The second half saw the battle largely confined to midfield although Wayne Curtis and Thompson went close with efforts for Morecambe and Jamie Pitman's sweetly struck drive narrowly cleared the Shrimps' bar as Hereford tried to add a second.
Blackburn, who had been booked for diving in the first period, was sent off following a foul on Jeannin seven minutes from time, but even with the additional man, Hereford were relieved to hear the final whistle.
"Under the circumstances, that's a big result for us as Morecambe will be up there at the end of the season," said Hereford boss Graham Turner.