Jake Jervis wasted Pompey's best first-half chance after 14 minutes when Ricky Holmes found him in space on the left side of the Cheltenham box, but he lashed his shot badly wide of the far post.
Jermaine McGlashan should have made it 1-0 for the visitors after 20 minutes when Nicky Shorey slipped as he tried to clear Byron Harrison's through ball. Pompey goalkeeper Trevor Carson leapt from his line and smothered McGlashan's effort.
From the resulting corner, the ball fell to Ashley Vincent 20 yards from goal and he lashed a stunning strike against the Pompey post.
Portsmouth dominated the second half, which saw Fogden's close-range effort smuggled wide of the Cheltenham post after 71 minutes, before Holmes broke free on the left four minutes later and his strike was deflected tantalisingly across goal.
Fogden shot wide of the Robins post with two minutes to play when the ball broke free after a goalmouth scramble as Cheltenham clung on for a point.