Winger Andy Robinson was outstanding for Simon Grayson's men who made it four successive home wins although they have suffered defeats at Huddersfield and Hereford United.
Jonathon Douglas returned after suspension for Leeds in place of Lee Trundle, while Andrew Hughes came in for Carl Dickinson who has returned to Stoke City.
Cheltenham, who had lost four games in a row, handed a debut to teenager Michail Antonio, on loan from Reading, while Nicholas Bignall was back after an eye injury. Striker Ashley Vincent and defender Shane Duff were out with ankle and back problems, respectively.
Leeds rapidly pinned their opponents back and Brown made successive saves from Howson and Luciano Becchio before Robert Snodgrass had a header blocked on the line.
Town, packing their midfield and leaving Lloyd Owusu as a lone striker, used Antonio's long throws to try and unsettle the home defence but to no avail.
Douglas headed against the bar when Town failed to clear a 27th-minute corner from Snodgrass and in the immediate aftermath Antonio raced away and shot fractionally wide of the right post.
However, the youngster, who had been hobbling since the early minutes, was replaced by Morley Watkins in the 29th minute and Leeds defender Hughes limped off four minutes later, Neil Kilkenny leaving the bench to replace him.
Brown remained defiant, using his legs to make a save from Snodgrass and showing lightning reactions to parry Neil Kilkenny's shot before Robinson fired against a post.
Leeds keeper Casper Ankergren had little to do but he made a brave save at Owusu's feet in a rare Cheltenham attack.
Brown was finally beaten in the 54th minute. Snodgrass released Robinson and his cross from the right was headed home by Howson from ten yards, much to Leeds' relief.
Bignall had the ball in the net for Cheltenham just after the hour, only to be denied by a flag, and Howson struck again in the 65th minute with a low shot from Fabian Delph's throughball.