Burnley forward Alex MacDonald marked the end of his loan stint with a superb free-kick to open the scoring before Constable's leveller in the 62nd minute after home keeper Rene Gilmartin flapped away a cross under pressure from Sean Rigg.
In the shoot-out U's keeper Ryan Clarke saved from Jamie Lowry and then Maxime Blanchard after Andres Gurrieri had missed his spot-kick to send Oxford through at the expense of Argyle.
Plymouth boss Carl Fletcher said: "At 1-0 we should have won it. It should have been done and dusted. It was a poor goal for us to give away from our point of view.
"We probably didn't play as well as we have done over the last few weeks but in the second half we had some good periods and some good chances.
"We had some good half chances and should have seen the game out really but we didn't and then you get to penalties and it's a lottery.
"They brought Constable on and changed their formation to try and get a bit more joy but I think we worked really hard at keeping them at bay so they only had minimal chances.
"We probably didn't keep our shape as well as we should have but once the rain started coming down it was very difficult and it was disappointing not to see the game out."
