The 23-year-old son of former Wigan and Castleford rugby league star Frano Botica kicked five penalties from five attempts for Quins, 31-30 winners at Northampton last weekend.
Botica's teenage replacement Louis Grimoldby added two second-half penalties to kill off a fightback by Bath, who had been comprehensively outplayed in the first period.
Former All Blacks fly-half Stephen Donald kicked three penalties for the visitors, with his replacement Tom Heathcote kicking one.
But their cause was not helped by the sin-binning of prop Anthony Perinese in the first half and replacement hooker Brett Sharman in the second.
Rob Miller's late penalty secured Sale Sharks' first triumph of this season's LV= Cup as they won 25-23 at the expense of Saracens at the Salford City Stadium.
Tries by Kameli Ratuvou and Joel Tomkins to go alongside 10 points from the boot of Nils Mordt gave Sarries a 20-12 half-time advantage, with Sale's response before the interval coming through Mark Cueto's score and a penalty try.
A Fraser McKenzie touchdown and a Danny Cipriani penalty took the Sharks into the lead before Ben Spencer's three-pointer edged the visitors ahead again, but Miller slotted a late penalty to earn the victory for the hosts.
- Sports & Recreation
- Frano Botica