Begovic grew up in Germany and Canada after his family fled Bosnia during the war.
Key players Asmir Begovic and Younes Kaboul were sold against Grant's wishes in January.
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"We have moved on and everything is well now, " the 23-year-old Begovic, who denied Pulis's claims, told BBC Sport.
Pompey have also tied down reserve goalkeeper Asmir Begovic until 2011 after the Bosnian-born 21-year-old agreed a new three-year-deal.
John Eustace had an opportunity at the other end but the experienced midfielder headed straight at on-loan goalkeeper Asmir Begovic.
"Things are going really well, " said Begovic, who had loan spells at Macclesfield, Yeovil, Bournemouth and Ipswich during his four years at Portsmouth.
Giggs was found by a Darren Fletcher ball from midfield and immediately curled in a ball for Rooney to slot past keeper Asmir Begovic.
It got worse for the Republic of Ireland international in the 62nd minute, as he again headed past Asmir Begovic from Juan Mata's corner.
Begovic and Pulis had a public falling out in August when the manager said the Bosnian had refused to play in a Carling Cup tie against Shrewsbury.
The hosts could have levelled in the 54th minute when Stoke goalkeeper Asmir Begovic flapped at Barrera's corner and James Tomkins put a diving header just wide.
Following the spat, Begovic had to wait until 27 October for a Premier League start, when an elbow injury to regular keeper Thomas Sorensen handed him his chance.
These included Stoke completing a deal for keeper Begovic on a four-and-a-half-year contract, striker David Nugent's loan deal being extended at Burnley, while Tottenham re-signed central defender Younes Kaboul.
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Pulis said Begovic had told him "he did not want to play" against the Shrews, though the Bosnian international retaliated by describing his manager's claims as an "absolutely disgraceful accusation".
He netted his third goal in the 54th minute after Piquionne was penalized for fouling Giggs, sending Asmir Begovic -- deputizing for the injured David James -- the wrong way.
Carlos Tevez fired wide from a free-kick and goalkeeper Asmir Begovic, starting in the place of injured England international David James, needed to recover quickly after fumbling from another set-piece.
Ellington had his first sight of goal 17 minutes from time, but the former Wigan striker failed to get any power on his long-range strike and Begovic made a comfortable save.
Potters goalkeeper Asmir Begovic comfortably saved from Gary Neville before Giggs slotted home with his first-time finish after a low cross from Berbatov was allowed to run across the Stoke area.
Sorensen returned to the team as one of eight changes for Saturday's 1-1 FA Cup draw with Cardiff, but Begovic is confident he can hold on to his place in Pulis's full-strength side.
The home side created chances of their own though to kill the game and thought they had done so when Begovic dropped the ball under pressure from Younes Kaboul and Pavlyuchenko hooked in.
Rooney completed his hat-trick - his third for the Old Trafford outfit - by sending Begovic the wrong way to give United some breathing space before Giggs curled a 22-yard free-kick into the bottom corner.
The financial problems have also impacted upon the hierarchy at the club, with chief executive Peter Storrie and manager Avram Grant claiming last week that they had been unaware of negotiations concerning the sale of Kaboul and Begovic.
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Shortly after, the home side doubled their lead and again Russian Pavlyuchenko was involved, holding up the ball before laying off to the onrushing Modric, who skipped past Huth and fired in low between Stoke keeper Asmir Begovic's legs.
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