Close's thumping volley and a home debut double from summer recruit Ellis Harrison helped Kenny Jackett's men advance to round two for the first time in four years.
Midfielder Close rifled past goalkeeper David Stockdale from 22 yards five minutes before half-time after Ronan Curtis' cross had been deflected into his path.
Pompey had earlier taken the lead when Harrison's close-range header had been adjudged to have crossed the line before Wes Harding cleared the ball onto his own crossbar.
Harrison grabbed his second and Pompey's third nine minutes into the second half as Birmingham paid the price for making nine changes from their opening day win at Brentford.
Curtis picked out the former Ipswich striker at the far post and Harrison nodded the ball back across goal and into the far corner.
Harrison would have been celebrating a hat-trick but for a smart save from Stockdale to keep out his close-range shot.
The only bright note for Birmingham fans was seeing 16-year-old Jude Bellingham break Trevor Francis' record as the club's youngest ever player.