Lukas Jutkiewicz's deflected strike gave Boro the lead after half an hour, but the Argentinian replied with goals in the 45th and 73rd minute to secure the points for the hosts at Elland Road.
Becchio now has seven in seven league and cup games, but Warnock insists he needs a new recruit to take some of the goalscoring burden off the striker's shoulders.
"Luciano's first goal was unbelievable," said Warnock. "It was not even a half chance but a top-quality finish.
"It was the first time he had ever been on the winning side in a five-a-side game in training on Friday. He looked so lively that I said 'is there any chance you'll be like that tomorrow' - and he was!
"I still feel we need another striker because Luciano can't play week in, week out, with so many games coming up and we need someone in case he gets injured. That has to be my priority.
"It was a super day for our new owners at the end of a really positive week. I have to be delighted, especially coming from behind to beat a very good side.
"We put our bodies on the line near the end and there were a lot of tired legs out there."