Somerset chose to bowl but were kept waiting for their first breakthrough as openers Lategan and Brett D’Oliveira (45) added 85 in 15 overs.
The bowlers gave little away though and as risks were taken in search of acceleration, wickets began to fall.
D’Oliveira drilled Kasey Aldridge to extra cover and Kashif Ali was caught with the greatest nonchalance by Finley Hill on the mid-wicket boundary off Green, while Jake Libby skied Aldridge to mid off.
Nineteen-year-old Lategan posted a polished maiden fifty but fell lbw, reverse-sweeping Leach as the spinner built pressure which also saw him bowl Rob Jones.
After Ethan Brookes was lbw, sweeping Tom Lammonby, Worcestershire needed some late-order contributions and they got them from Henry Cullen (32 off 30) and Matthew Waite (24 off 26) before Tom Taylor clubbed three sixes in an unbeaten 22 off 10 balls.
Pakistan international Shahzad then began with two wides, one of which went for four, in his first three balls.
But Lammonby and Lewis Goldsworthy were beaten for pace when trying to pull and sent up catches, Archie Vaughan was brilliantly caught by Ben Allison at mid-wicket and Thomas Rew edged to slip.
Worcestershire’s other seamers deepened Somerset’s plight. Waite had Joshua Thomas well-caught by Lategan on the long leg boundary and Ben Allison dismissed Hill, caught at slip, and Green, at mid off.
Green fell to the last ball before heavy rain arrived with Somerset in dire straits.
They resumed needing another 165 from 14.2 overs to reach a revised target and quickly lost Aldridge, caught behind, and Leach, stumped, off Brookes.
Last pair James Rew and Jake Ball needed to find 137 runs from 10 overs but they only managed five before Rew lifted Brookes (3-16) to long off to trigger loud and long celebrations from the home fans.
ECB Reporters’ Network supported by Rothesay.