The 19-year-old joined on loan from West Ham on Thursday and scored in both halves before James Walker got the third. Michael Spillane got the Hatters' goal consolation goal.
Both sides had needed replays to overcome non-league opposition in the first round and there was little to get excited about in the first half.
Southend's debutant winger Stanislas, on loan from West Ham United, curled a free-kick into the sidenetting, before Southend keeper Steve Mildenhall comfortably dealt with tame efforts from lone striker Chris Martin and Paul McVeigh.
The Shrimpers upped the pressure with Alex Revell's shot being turned out for a corner by Luton goalkeeper Conrad Logan and then Francis Laurent's ten-yard shot was blocked by Spillane.
The Blues then opened the scoring following a controversial free-kick. Assistant referee Marvin Thompson raised his flag for offside but referee Andy Penn ignored it and booked Ian Roper for a challenge on Kevin Betsy, while the assistant lowered his flag.
From the dead ball, Stanislas cracked a 25-yard shot into the top right-hand corner of the net with Logan rooted to the spot.
Southend stepped up the pressure but were unable to force a second goal through poor finishing and some dogged Hatters defending.
Luton were happy to counterattack but had little to show for it apart from an off-target effort from Asa Hall.
Johnny Herd and Laurent were both denied by Logan before the Hatters got on terms ten minutes from time.
McVeigh curled in a corner from the right and Spillane headed the ball goalwards. Laurent looked to have made a goal-line clearance but assistant referee Thompson signalled for a goal.
But the period of parity lasted just four minutes before Betsy's cross from the right was headed into the top left corner of Logan's goal by Stanislas from eight yards.
Southend wrapped up the win in the 90th minute when substitute Anthony Grant burst into the box and teed up fellow sub Walker, who smashed the ball home from 12 yards.
A bit of Shrimpers history was made in stoppage time when 15-year-old Femi Orenuja became the youngest player to feature for Southend's first-team.