James did enough to put off Nemanja Vidic as he rose to meet a free-kick and the veteran goalkeeper also denied Giggs twice in quick succession.
Another glorious chance came and went when Nani stole possession from Johnson and crossed for Fletcher, who was unmarked in the middle but failed to connect with his shot and knocked the ball away from goal with his second touch.
By now Campbell was having a torrid time and he was embarrassed again when Nani cleverly chipped him and also clipped the ball over James before Sylvain Distin cleared the danger.
Pompey created next to nothing in the first half, with their £18.5m strike partnership of Peter Crouch and Jermain Defoe barely having a sniff of the ball, let alone a meaningful chance.
And United stayed well on top in the second half with Ferguson's side continuing to pour forward.
Diarra blasted Portsmouth's first spot-kick over the bar
Fletcher almost made amends for his earlier miss when he picked up James' punched clearance and lobbed the ball goalwards only to see the keeper somehow recover to turn the ball on to the crossbar.
Giggs was the next player to waste an opportunity when he scuffed his shot in front of goal, before James produced another wonder-stop to push Tevez's curling effort around the post.
Portsmouth were fortunate again soon after when Hermann Hreidarsson grabbed the Argentine's leg and hauled him down in the area but referee Peter Walton ignored the challenge and then missed Tevez's reaction - which saw him grab the defender by the neck.
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It took 76 minutes for Harry Redknapp's side to muster a shot on target - a looping effort by substitute John Utaka that presented an easy catch for Edwin van der Sar.
But United's inability to find the net meant Pompey took the game to a shoot-out, only to fluff their best chance of victory.
Lassana Diarra set the precedent when he blasted their first spot-kick high into the stands and, although Defoe converted his effort, Arnold Mvuemba saw his shot saved before Johnson missed to hand United victory.
Tevez, Giggs and Michael Carrick all made no mistake for United who pick up the first piece of silverware of the new season.
Man Utd: Van der Sar, Neville (Brown 66), Ferdinand, Vidic, Evra, Fletcher, O'Shea (Carrick 66), Scholes, Giggs, Nani (Campbell 79), Tevez.
Subs Not Used: Kuszczak, Evans, Possebon, Da Silva.
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