Yes, that's exactly what he should have done. The risk of fouling the player was minimal as the ball was easily there to be won; the challenge would have had to have been very wild to have fouled the player, if de Gea had have attacked the ball with conviction he'd have cleared the danger with very little risk at all.
A second, though somewhat less preferable option would have been to stand his ground and make himself a big target; Rodriguez would have had much to do in such a situation, and once he'd taken a touch de Gea would be free from any pass-back restrictions.
Either of the two options described above would have been preferable to what he did in fact do, which was absolutely the wrong thing to do. He made no effort to either win or block the ball, he completely pussied out of the challenge, clearly caught between two minds and not really having a clue what to do. He chickened out, jumped up a bit and spun sideways, in the process making himself probably as skinny and small a target as was possible in the situation; this was incredibly weak goalkeeping, there can be no argument otherwise.
The description from the Guardian is perfectly accurate. Michael Carrick was at fault for putting de Gea in a difficult situation, but de Gea was certainly at fault for handling that difficult situation in the way he did; both players played their part in gifting Southampton the goal.
I've defended de Gea from much bullshit in this thread, I think his coverage in the press is unfairly critical, but you're an idiot if you genuinely think he didn't make a big mistake in how he handled that back-pass last night. Sometimes, as a football supporter, one can defend or attack a certain player's merits to the extent that one becomes completely blind to any evidence that might contradict one's own opinion, and thus, with so many of you unable to see how de Gea was at fault for the goal last night, it seems that this is exactly what is happening here.
De Gea made a big blunder at the start of the game against Southampton which led to us conceding a goal; he wasn't the only player at fault, but he was certainly at fault. You need to deal with that, because to argue otherwise makes you look like a Ihni binni dimi diniwiny anitaime.