It was the perfect start from the visitors as they took the lead inside two minutes when Ionut Nedelcearu curled home a superb free kick.
Romania were dealt a huge blow just before the interval when goalkeeper Laurentiu Branescu foolishly kicked out at Tommy O'Sullivan after being fouled, which gave the referee no other option than to show him a straight red card.
O'Sullivan twice wasted good opportunities early in the second half as he shot wide while sliding to reach a cross, before sending a half-volley just a few inches wide of the post moments later.
Burns continued to look the most dangerous player on the pitch, and he almost handed Wales the lead as he turned past a defender before rattling the post.
The away side continued to defend impressively with 10 men on the pitch and they dropped deep to frustrate the hosts for much of the second half.
A predictable onslaught arrived in the closing stages as Wales searched for a winner, with Burns heading over late on, but they were forced to settle for a point which keeps them top of the group.