But the fly-half missed a second penalty and a drop goal in quick succession as the Warriors failed to capitalise on their early possession.
And it looked like they would pay for it when Grewcock lunged for the line from close range, but Benjamin's initial tackle stalled the lock's progress and Tuitupou's follow-up knocked the ball clear.
But, after Walker and Davis had traded a couple of penalties each, Worcester's hopes of snatching victory from 12-9 down looked over when Tuitupou was shown the yellow card for leading with his shoulder into a tackle.
If anything, the referee's decision reinvigorated Worcester who pressed in the last 15 minutes, no doubt conscious of the fact that Bath had let leads slip late on against Newcastle and Harlequins last month.
And their efforts paid off when Walker took his chance to clinch a draw.
Worcester coach Mike Ruddock:
"If it was a boxing match I think we would probably have won on points.
"They were hard to break down and sharp on the break.
"We couldn't find the killer blow to put them away. We had a couple of purple patches but we didn't take advantage - we had to be more clinical."
Bath coach Steve Meehan:
"We had a couple of chances with six minutes to go. We should have drilled it into the corners and then we would have had the chance to win it.
"However, without looking at the stats, it appeared to me that we spent a lot of time in our half under pressure.
"We could do with a bit more creativity and we will have to work on that."
Worcester: Latham; Garvey, Rasmussen, Tuitupou, Benjamin; Walker, Powell; C Black, Lutui, Taumoepeau, Rawlinson, Gillies, Wood, Sanderson, Horstmann. Replacements: Gear for Garvey (55), Arr for Powell (64), A Black for C Black (68), Fortey for Lutui (68), Sourgens for Taumoepeau (68). Not Used: Kitchener, Abbott, MacRae.
Sin Bin: Tuitupou (66).
Bath: Abendanon; Carraro, Hape, Williams, Cuthbert; Davis, Claassens; Flatman, Dixon, Jarvis, Short, Grewcock, Skirving, Salvi, Watson. Replacements: Stephenson for Williams (57), Catt for Flatman (71), Mears for Dixon (60). Not Used: Lilley, Hobson, Browne, Bemand, Roberts.
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