Alex MacDonald, who will now return to Burnley following the end of his loan stay in Devon, curled in a scorching free-kick from the left-hand touchline in the 20th minute to give the home side the lead, an right-footed effort that gave Clarke little chance.
Oxford equalised in the 62nd minute through target man James Constable after home keeper Rene Gilmartin flapped away a cross under pressure from Sean Rigg. The ball spun to 55th minute substitute Constable, who looped his header over Gilmartin, and in.
In an open quarter final tie, watched by 2383, Plymouth right back Durrell Berry smashed a 20-yard shot off a post in the 16th minute, not long after Gilmartin had to race out of goal to clear from Oxford attacker Rigg.
Plymouth were reduced to 10 men when Berry received his second yellow card for with six minutes to go before the shoot-out ensued.
Oxford slotted in all three of their penalties with Rigg, Adam Chapman and then Liam Davis all finding the back of the net. But after Rhys Griffiths scored the first penalty of the shoot-out Argyle trio Andres Gurrieri, Jamie Lowry and finally Maxime Blanchard all failed to convert from 12 yards.