The midfielder rolled the ball into an empty net to cap a fine 76th-minute move from the hosts after Stevenage had threatened to cause an upset thanks to Christophe Berra’s first-half own goal.
Youngster Josh Yorwerth powered home an equaliser in the 55th minute before Tabb grabbed the winner.
The game ended in controversy when the visitors thought they had been awarded a controversial 83rd-minute penalty – the ball was even placed in the spot – but referee Fred Graham overruled his assistant’s initial verdict.
A scrappy first half saw Stevenage take a surprise lead 11 minutes before the interval when the unfortunate Berra deflected the ball into his own net after keeper Dean Gerken had saved from Dean Parrett.
Ipswich responded well with Brett Pitman and Tabb going close before the interval before the equaliser finally came 10 minutes after the restart thanks to Yorwerth’s downward header.
Strike duo Pitman and David McGoldrick then had second-half headers saved by veteran keeper Chris Day before he was finally beaten when McGoldrick rolled the ball across to Tabb to tap into the unguarded net.