Andy Robinson gave the home side the lead with a superb 74th minute free-kick, but after both sides were reduced to ten men, Rocky Baptiste struck an 86th minute equaliser to give the Hawks a deserved draw.
There were a number of changes in the Swansea line-up and Alan Tate took over the captain's armband from Gary Monk who was on the bench.
The Swans attacked from the opening whistle and after a Joe Allen centre zipped across the danger area without anyone getting a touch, Robinson made space before striking his shot wide.
Havant worked the ball down the left flank in the 13th minute but skipper Jamie Collins fired straight at home keeper Dorus de Vries.
Five minutes later, Leon Britton started and finished a move by rattling the crossbar as the League One leaders looked for the goal they needed to settle themselves down.
Despite Swansea's superiority, a poor final pass often let them down and play became disjointed.
When Phil Warner brought down Leon Britton in the 42nd minute, Robinson's free-kick was headed wide by Tate from a good position.
Yet another free-kick for the Swans saw Robinson's curler drift just wide of the far post.
Only a block by Justin Gregory on the hour prevented the Swans going ahead as Jason Scotland shot for goal, but the home side missed another chance soon after when Darren Pratley thundered a shot against the woodwork.
The breakthrough finally came when Robinson netted from a free-kick just as he had done the previous week against Leeds.
A full-scale battle ensued in the centre of the field after Brett Poate's two-footed tackle on Andrea Orlandi he was immediately red-carded.
But as soon as referee Stuart Atwell regained control, he dismissed Swansea skipper Tate and booked home defender Kevin Austin and Havant's midfielder Shaun Wilkinson.
Four minutes from time the home defence failed to clear a Tony Taggart centre it was flicked on by Alfie Potter and Baptiste was waiting at the right-hand post to fire in a deserved equaliser.
The Swans had chances to regain the lead in the last few minutes but the Blue Square South side hung on for a well-deserved draw.