Former Shrimpers striker Billy Paynter put Swindon ahead with just 45 seconds on the clock when he fired low into the right-hand corner. Francis Laurent then hit the bar for Southend before the visitors added a second in the 10th minute.
Baldwin's clearance hit Simon Ferry and the ball fell to Charlie Austin, who fired into the bottom right corner.
Southend pulled a goal back in the 85th minute when Simon Francis' free-kick produced a one-handed save from Phil Smith and substitute Spencer stabbed home the ball in the resulting goalmouth scramble.
And Baldwin, making his debut after arriving on loan from Essex rivals Colchester, secured a share of the spoils with one minute of normal time remaining when he headed Damian Scannell's cross into the centre of the goal from close range.