Town took the lead on the half-hour and home keeper Jan Budtz made some great saves as they swept forward against an out-of-sorts home team, despite losing skipper Hasney Aljofree through injury with only seven minutes on the clock.
But one goal was all they could muster despite their domination and that gave Pools hope in the second half, when two subs and some tactical changes gave them renewed energy.
Three Swindon new boys combined for their goal, Simon Cox setting up Billy Paynter to cross from the right and Jon-Paul McGovern controlled it well in the middle before chipping the ball over Budtz as Hartlepool protested it had gone out for a throw before the cross.
But in the second half it was a different story as Pools took command and on 69 minutes sub Joel Porter blasted them level, cutting inside from the right before planting a low left-foot shot into the bottom corner from the edge of the area.
Swindon threatened on the break and Budtz made a brilliant one-handed save from sub Chris Blackburn, before the game ended in drama when James Brown burst into the Town box and seemed to be brought down by Jerel Ifil.
But despite the home protests, the penalty appeals were turned down and a draw was probably a fair result over the 90 minutes after the teams took it in turn to dominate.
Both sides were promoted from League Two last season and on this showing neither should have anything to fear in League One.
If either or both can find some better finishing, they might even have a chance to repeat their promotion success.