Lallana took only eight minutes to give the League One side the lead, sweeping home from eight yards from Ryan Dickson's low cross.
The highly-rated England under-21 midfielder turned provider five minutes into the second half by firing in the centre from which Brazilian Guly Do Prado headed Saints' second.
Teenager Oscar Gobern completed the rout in the dying minutes with a looping far post header after Lallana's corner had been flicked on by captain Dean Hammond.
League Two Cheltenham rarely looked capable of causing an upset and David Bird's scuffed shot wide from home goalkeeper Bartosz Bialkowski's misplaced clearance was as close as they came to scoring.
Saints - cup winners in 1976 - would have made the victory even more emphatic but for Do Prado's first half header from point blank range rebounding off a post.