Mark Robins' men were a goal to the good twice through Wright and Milan van Ewijk, but were pulled back first by Ciaran Brown and then Mark Harris before the American sealed all three points at the CBS Arena.
Wright was the star man for the Sky Blues as he opened the scoring after 15 minutes when he powered home a header following Van Ewijk's inviting cross from the right.
"I thought Haji was brilliant all night," said City manager Mark Robins. "I think the team played really well apart from the two goals, they were really avoidable and really poor, but if we can win a game and concede those goals then fine tonight, but we will come unstuck if we can't defend and keep clean sheets.
"It's a relief and I've said it in years gone by, but this year might be the year I have a heart attack if we're going to play like that."
Oxford manager Des Buckingham admitted football was a cruel game as his side suffered their first loss back after 25 years in the English second tier.
"Football is a cruel game, if anyone was trying to show a clip of what a cruel game looks like, it's that," said Buckingham.
"As frustrating as it is, I've just said to the players 'I couldn't be prouder of what I've seen over two games back in the Championship. If we stick true to that over the next 44, we'll put ourselves in the best position to get more results'".
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