Twiss made it 17 league goals for the campaign with a brace that earned Jim Harvey's side the three points and kept their play-off hopes alive.
Former Chester striker Twiss kept his nerve to give Morecambe what could be a vital victory when he slotted home an 86th minute penalty after Bulls defender Tom Smith was adjudged to have brought down Danny Carlton in the box.
The game only really came to life after Danny Williams had given Graham Turner's side a 61st-minute opener.
The midfielder got the finishing touch after some great work on the left wing from Lee Mills and the visitors looked a good bet to take the points.
But Morecambe, unbeaten at home in seven games, staged a quick revival. Bulls keeper Craig Mawson saved superbly from Wayne Curtis before Twiss opened his account with a smart volley after 66 minutes.
Hereford started brightly with Danny Carey-Bertram soon in the thick of the action.
After three minutes he had a good run and shot from the edge of the area that went just wide before volleying into the side netting two minutes later after latching on to a fine cross from Jamie Pitman.
The best effort of the first half, however, came from veteran striker Mills in the 34th minute. Morecambe centre-back Darren Kempson misjudged a long clearance, gifting Mills time and space on the edge of the box but his shot was well saved by Ryan Robinson low to his left.
At the other end, Hereford's former Shrimps keeper Mawson reacted well to save from Curtis' 14th-minute effort but could nothing to foil Twiss' late double.