South Australia paceman Brendan Doggett, also in the 15-man Ashes squad for Perth, showcased why he is regarded as next cab off the rank for Australia with 5-66 in a superb exhibition of seam bowling on a green Bellerive Oval deck.
Weatherald, hoping to make his Test debut as an opener, fell for 23 in Tasmania's total of 209 after being sent in to bat.
Allrounder Webster has been superb in his opening seven Tests for Australia but his desire to hang onto his No.6 position in Perth next week was not enhanced by his knock of 13.
The 31-year-old missed a month of cricket at the start of the summer with a calf injury and made just 11 and six in his only other Shield clash this season.
His chances of featuring in the Australia side for the first Test in Perth could depend on whether selectors opt to bat Weatherald or Marnus Labuschagne as an opener. If it is the latter then Webster would stay at No.6 with fellow allrounder Cameron Green likely slotting in at No.3.
Weatherald dispatched Liam Scott for back-to-back boundaries and was aiming for a third but chopped an attempted cut shot onto his stumps.
Webster had just struck a fierce pull to the boundary before attempting to hook Henry Thornton. He only succeeded in getting a feather through to Alex Carey.
Tasmania skipper Jordan Silk brought up his 6000th Shield run in his innings of 64.
Silk was severe through the off side and brought up his 50 with a rollicking slap through point off Nathan McAndrew straight after he had caressed a cover drive to the boundary.
Doggett dismissed Silk with a full delivery that moved off the seam and was brilliantly caught by a diving Carey. He then ripped through the lower order for his second five-wicket haul of the Shield season.
South Australia were 3-88 at stumps. Australia No.5 batter Travis Head made just nine before falling to Webster.
Carey had raced to an unbeaten 25 off just 20 balls and will resume on Tuesday alongside Henry Hunt (34no).
Australian chairman of selectors George Bailey said there was "still some information to gather" from the fourth round of the Sheffield Shield before settling on the playing 11 against England.
Bailey said Weatherald, who was aggressive in his knock on Monday, had the approach Australia's brains trust admired.
"There is a method (about his batting) that we like and that is complementary to the other players around him in the squad," Bailey said.
"We have been chatting a lot to Jake. He has been in quite a few (Australia) A teams and there has been a lot of communication with him throughout the summer. The feedback has always been to keep doing it his way.
"That is what is attractive to us, that he scores at a nice rate and puts bad balls away. He is aggressive with his outlook without being over the top."
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