Marcus Rashford's penalty proved the difference between the two sides in the friendly but Southgate was left underwhelmed by his team's performance with just a week until their first game of the tournament against Croatia.
The England boss told Sky Sports: "I don't think we were as good as we want to be, or that we need to be.
"In the first half we used the ball reasonably well, our best moments were coming down the left, we needed to get the ball from right to left more urgently.
"We were a bit slow moving the ball, and in the final third we didn't quite select the right pass or ran into traffic when we had a chance to create clearer opportunities.
"We didn't do enough without the ball in the first half, especially our pressing from the front. That meant we were stretched, and we have to learn from that. We'll play against better teams who will carve us apart; Romania created more opportunities than we would have liked, and gave our goalkeeper a nice debut in that he had a bit more to do than we wanted to see."
Jordan Henderson squandered the opportunity to double England's lead by missing another penalty and Southgate was irked that the Liverpool captain took the penalty.
The Three Lions boss said: "We had Rashford on penalties and then Ward-Prowse, they'd both come off.
"We were expecting Dominic to take it, I was expecting Henderson to walk over and make sure he took it - so I'll see what his thought process was there. We'll pull rank on him over that."