Levi Landricombe gave Tivvy a 27th minute lead but it wasn`t until the 63rd minute that they were able to double that, courtesy of Owen Howe.
Landricombe made the points safe with his second and Tivvy`s third on 73 minutes, with Jack Longford netting a late consolation for Diddy.
Bridgwater Town and Wantage Town are both still looking for opening wins.
Bridgwater lost their sixth game out of seven – and comprehensively too. Evesham United hit five second half goals without reply at Fairfax Park.
And Lance Smith grabbed four of them in the 55th, 80th from the penalty spot, 88th and 90th minutes – the other coming from the home side`s Elliot Nelson past his own keeper in the 68th minute.
Wantage lost 2-0 at home to Yate Town, whose Jake Ball after 22 minutes and Kye Mountford 11 minutes from time earned a second win.
Newcomers Barnstaple Town`s Ryan Turner grabbed the only goal of the game at Larkhall Athletic in the 65th minute, while relegated Bideford won for a fourth time in eight at home to Bishops Cleeve.
Two first half goals from Sean Downing were enough to give Biddy the points.
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