Centre-back Dunk broke the deadlock in the 79th minute following a cross by Inigo Calderon before Mackail-Smith sealed the south-coast side's victory from close range in the 90th minute.
Hyypia won the trophy twice as a Liverpool player but the Finn's Brighton team were made to sweat by a robust Cheltenham who, with better finishing, could have taken the game to extra-time.
Brighton, who have reached the second phase of the Capital One Cup for just the second time in six seasons, threatened first when left-back Adam Chicksen hit a right-foot effort from 20 yards.
Summer recruit Chris O'Grady should have hit the target when through on goal and with only goalkeeper Trevor Carson to beat as Brighton continued to threaten.
Cheltenham's stubborn resistance was eventually broken, however, when Calderon crossed to Dunk following a short corner by Kazenga LuaLua 11 minutes before full-time .
Mackail-Smith, who added spark to the Brighton attack following the opening-day setback to Sheffield Wednesday, completed the home side's victory with an unchallenged finish from 10 yards in the last minute.