Bath's pack dominated the game and Newcastle did themselves no favours by conceding a string of penalties, mainly at the breakdown, and an early try from Matt Carraro from which they never really recovered.
The Falcons conceded the opening try after just eight minutes when Vesty sliced through the Falcons midfield and delivered a perfectly-timed pass for Carraro to score under the posts. Vesty made it 7-0 with the conversion.
It took Newcastle a while to get going before a catch and drive and a crossfield kick from Jimmy Gopperth stretched Bath and a penalty resulted, which Gopperth landed in the 12th minute for 7-3.
Ben Williams banged through on the crash ball enough to put Newcastle on the back foot and Vesty clipped over a 30-metre drop goal for a 10-3 lead in the 15th minute.
Back came Newcastle, who now looked far more lively and some steady forward drives led to a penalty for offside and Gopperth made it 10-6 in the 18th minute.
He missed out with a 55-metre shot shortly afterwards and Vesty also missed much simpler chance with both sides straying offside in a tight midfield struggle.
But he did stretch Bath's lead to 13-6 with a penalty in the 33rd minute.
Newcastle responded strongly and Alex Tait and Chris Pilgrim both went close in a searing attacks, the Falcons having to settle for a penalty which Gopperth knocked over for 13-9 in the 37th minute.
Gopperth had another kick from 50 metres right on half-time but he was no more successful this time than he had been earlier from 55 metres.
Again Newcastle failed to look after the ball at the start of the second half and compounded the error by giving away another penalty for going in at the side.
Vesty easily made it 16-9 and then 19-9 after Newcastle were penalised again at the scrum in the 57th minute.
And it was Vesty who put the game beyond Newcastle's reach with his fifth penalty in the 63rd minute as the Falcons struggled to break the Bath forwards' stranglehold on the match.
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