The Seagulls were hit by a double salvo just before half-time in Saturday's League One clash at Elland Road.
First Becchio fired past Michel Kuipers from eight yards at the second attempt in the 45th minute.
Then, in the second minute of injury time, 16-year-old left-back Aidan White darted down the left and Fabian Delph's cross brought a fine header from Beckford.
The Whites striker scored his second of the match and his 13th of the season in the 90th minute when put through by Delph.
There was still time for Glenn Murray to fire home a consolation goal from 12 yards for Brighton just before the final whistle.
The result takes Leeds up to third, while Brighton are 17th.