But Gary McAllister's team made hard work of the points after spurning a hatful of chances at Whaddon Road with Jermaine Beckford the chief culprit.
The visitors, who had lost their last four away games, scored the decisive goal when Argentine hitman Becchio headed in a Frazer Richardson cross after seven minutes.
Cheltenham, with just one fit striker, struggled to carve out any clear-cut openings but Scott Murray spurned a good chance in first-half stoppage-time, drilling his shot just wide.
Paul Connor missed an even better chance at the start of the second half, shooting straight at goalkeeper Casper Ankergren from 12 yards with the goal at his mercy, while Josh Payne forced a finger-tip save from the Leeds stopper with a right-footed shot from 25 yards.
Beckford rattled the side-netting after the break and was denied a brilliant solo goal after Shane Higgs pushed his shot wide following a jinking run into the box.