Becchio, a second-half replacement for Davide Somma, headed home a pinpoint cross from Robert Snodgrass into the bottom corner to put the home side ahead.
But the Robins have built impressive momentum in recent weeks and lifted themselves to hit back immediately.
Albert Adomah's cross from the right caught the Leeds defence flat-footed and Jonathan Stead swept the ball home from six yards.
Back came Leeds again and in their very next attack they regained the lead. Lloyd Sam let fly from 20 yards and although David James got his hands to the ball, he could only parry it into Becchio's path and he slammed home the ball from close range.
The 26-year-old completed his hat-trick in the 83rd minute, powering home another header from George McCartney's left-wing cross.
Leeds goalkeeper Kasper Schmeichel brilliantly tipped Steven Caulker's stoppage-time header over the crossbar and then pulled off another superb save to deny City substitute Brett Pitman but the hosts were already home and dry.
Source: PA
