Alfie Pavey's penalty put the home side in front before Nathan Thompson levelled in the relentless summer sunshine.
Pavey was the first to aim a shot on goal with a speculative 45 yarder after five minutes, which flew high and wide.
Pompey's first effort arrived after eight minutes when Tom Naylor took aim from 20 yards, but his low drive went beyond the far post.
Three minutes later Brett Pitman and Ronan Curtis combined to give the Irishman a sight of goal 12 yards out, but Jordan Rose got a last-ditch tackle in.
Lee Brown then fired a shot at Ryan Young after 16 minutes, as Pompey began to exert some control of the game.
Craig MacGillivray then left his box two minutes late and won two headers, but the ball dropped to Brian Stock in the centre circle. Fortunately for the Pompey keeper the midfielder's flighted effort was off target.
Seconds later, a scramble ensued in the Hawks box from Gareth Evans' corner but no blue shirt could force the ball home.
Nathan Thompson was down hurt on the turf after a robust challenge from Jordan Rose, and was okay to continue after gingerly getting to his feet.
Burgess' shot was deflected seconds later as Pompey continued to threaten from a corner.
Hawks were given the chance to take the lead in the 29th minute as Matt Clarke was adjudged to have clipped the heels of Nicke Kabamba, as he burst clear.
The assistant referee flagged for the decision and Pavey sent MacGillivray the wrong way from the spot.
Pompey were level in the 34th minute when Thompson found space in the box to angle a low drive past Young
Pitman converted a ball over the top six minutes before the break, but the offside flag was correctly raised.
Lowe's shot from 12 yards cannoned off Paul Robinson's head and away to safety in the last action of the half.