An 18th-minute header from Scott Hogan was enough to secure a win for Steve Bruce's team, who were far more comfortable than the margin of victory suggested.
Sam Johnstone only had one meaningful save to make in the Villa goal, as Kieran Dowell tested him late on. But that was as close as the home side came to an equaliser, as an organised, resolute defence, marshalled well by John Terry, largely held firm.
Former Boro boss Karanka, who began work on Wednesday after succeeding Mark Warburton, believes his new club are capable of better than they demonstrated on a frustrating evening.
"This was our first game and, for sure, we need to improve. There were positive things, but obviously not as many as we would like," he said. "This was our first game and there was a lot of expectation, following the win over Arsenal. But Villa are a very experienced team.
"We have a lot of potential, but we are young and we need to improve. I saw the quality we have. We are young and we can improve a lot. But we have to be more aggressive and intense.
"We have not had time to prepare for the game in the way that I like to. But now we have one week to do everything and to keep encouraging the players, because they need our support.
"They are a young group of players and they need to be helped. We have a long way to go, if we want success. We need to improve."