Pontypridd were on the back foot for the most part in a full-blooded encounter, although when they reshuffled their pack they forced Bath into desperate defence.
The spirit and resilience of Bath was epitomised by Phil de Glanville, who went off with blood streaming from a head wound but returned to pull off a series of crucial tackles.
Pontypridd played by far the more expansive game but in the first half their scrum was in serious trouble against Dave Hilton, Mark Regan and Victor Ubogu and they had nobody to match the athleticism of the American, Dan Lyle, in the lineout.
Bath were soundly beaten in the Cup here last season and Robinson said: "We knew what to expect and we built ourselves up. Our attitude, focus and the absolute physical nature of our game was the best it's ever been. In the first 60 minutes it was very much the Bath of old."
Pontypridd had conceded two tries - they were furious about the legality of the first - and were 18-3 down before they belatedly made tactical changes that brought them back into the game.
In the first minute Gareth Wyatt crossed the Bath line but was adjudged to have put a foot in touch, and then a stupid late tackle by Dale McIntosh on Jon Callard resulted in a penalty for the full-back.
The first try came in the 30th minute when the impressive Richard Butland was the first to reach a chip by Mike Catt. Pontypridd claimed the ball was over the dead-ball line and although it look desperately close Mr McHugh was perfectly placed.
Bath, who led 10-3 at half-time, went further ahead in the 45th minute when Lyle broke from a scrum, Callard cut inside and Ubogu crashed over.
The backlash began when the Welsh champions changed their front row and almost immediately Dafydd James finished off a smart attack. When Steele Lewis added another, Bath's lead was cut to six points and they had to endure a torrid final quarter.
At one point Robinson, in his enthusiasm to issue instructions to his team, was ordered from the touchline. He got round that by using the Bath physiotherapist as a messenger. Bath may have had more than their fair share of luck, but nobody could doubt their professionalism.
Pontypridd: Tries James, S Lewis; Conversion Jenkins; Penalty Jenkins. Bath: Tries Butland, Ubogu; Conversion Callard; Penalties Callard 3.
Pontypridd: K Morgan; G Wyatt, J Lewis, S Lewis, D James; N Jenkins (capt), P John; N Eynon, J Evans (P John, 54), A Metcalfe (M Griffiths, 52), G Prosser, M Rowley (S Roy, 55), M Spiller (G Lewis, 75), D McIntosh, M Williams.
Bath: J Callard (capt); M Perry, P de Glanville, M Catt, B Roche; R Butland, R Pellow; D Hilton, M Regan, V Ubogu, G Llanes (M Haag, 64), N Redman, R Webster, D Lyle, E Peters.
Referee: D McHugh (Ireland).Reuse content