After slotting home a late spot-kick to give the Shrimps three points against Hereford he repeated the feat with a similar effort to earn Jim Harvey's side a vital point from a thrilling encounter against Exeter City.
It was Twiss himself who was brought down by a clumsy tackle by Jake Edwards, seconds after he had forced Paul Jones into a wonderful save with a close-range header.
The penalty sparked an action-packed finale to a well-contested game and Morecambe came desperately close to grabbing all three points in the fourth minute of injury time.
After more good work from the dangerous Twiss, Garry Hunter was presented with a golden chance but although his right-foot shot beat Jones, veteran defender Scott Hiley was perfectly positioned to head the ball off the line.
The game itself came to life after Exeter took the lead on 40 minutes. Alex Jeannin's cross found Chris Todd and his shot across goal was tapped in by Lee Phillips at the far post.
As they did in midweek against Hereford, Morecambe bounced back almost immediately with a goal five minutes later.
Michael Howard broke up an Exeter attack and played a neat throughball for Danny Carlton. The young striker teed up Garry Thompson whose shot from an acute angle hit the post but fell perfectly for Carlton to slam home the rebound.
The Shrimps should have gone 2-1 ahead just seconds after the restart. Twiss shared a neat one-two with Wayne Curtis with the latter put through on goal, but his effort was somehow blocked by the feet of Jones.
The Grecians regained the lead after 53 minutes when Jeannin fired home from the edge of the area and looked to be on their way to a vital victory.
However, City had to settle for a point after Twiss kept his cool from the spot three minutes from time.