Leicestershire started the day with a lead of 60 but started slowly as Glamorgan were content with a holding operation in the field, Kiran Carlson and Andy Gorvin keeping it tight.
Tattersall reached his first Leicestershire century, in his second game since moving from Yorkshire, off 191 balls and including just six fours.
But he quickly fell, caught off Carlson, when he tried to open his shoulders having put on 104 with Ben Cox.
Cox recorded his third score of 50-plus in consecutive innings, including 162 against Surrey, but although Ben Green fell quickly swinging at Gorvin, there was no concerted flurry of shots before the declaration at 12:32 BST with 69 overs left.
Australian opener Jake Weatherald did not resume his innings for Leicestershire after suffering cramp and retiring on 104, with the team not needing his batting services or a permanent replacement in the match.
Eddie Byrom’s scratchy form continued as he fell lbw to Josh Davey for five, but Zain Ul Hassan patiently soaked up 80 balls for 29 before playing no shot to Ben Green and chopping on to his stumps.
But eight wickets in the final session looked a tall order for the Foxes bowlers and so it proved, despite a fierce spell of short-pitched bowling from paceman Josh Hull which saw Carlson struck painfully in the ribs.
Tribe was largely untroubled after surviving some early difficulty against the spin of Ajaz Patel, while Carlson finished on 28 not out as very occasional spinner Stevie Eskinazi got a chance to turn his arm over along with Rishi Patel.
Glamorgan and Leicestershire both have two draws and a defeat from their opening three games back in the top flight, with the Foxes extending the points gap between them to five.
Leicestershire host champions Nottinghamshire in the next round starting on Friday 1 May, while on the same day Glamorgan begin a rare Championship visit to Hampshire.
