Becchio hoisted his tally for the season to 17 and his recent scoring spree to seven in seven league and cup games - but an additional striker is still on Warnock's wish-list.
Warnock said: "Luciano's first goal was unbelievable. 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."
Middlesbrough had led through Lukas Jutkiewicz's deflected strike but manager Tony Mowbray said: "We had a host of chances and missed them.
"We were through one-on-one two or three times and threatened the goal more than enough to have got something out of the game, but give Leeds credit.
"They worked hard and defended well without putting us under much pressure. They had a tough game against Chelsea on Wednesday and sent just about the same players out against us. They got a result so good luck to them.
"Becchio's first goal was a great one. To turn and hit the ball as well as that showed great technique, but our own finishing needed to be more clinical."
- Sports & Recreation