The 19-year-old Staniskas 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.
Speaking to Blues World after the game Blues boss Steve Tilson said: "We should have been out of sight by the time Luton scored. I was worrying a bit and you start to think, 'here we go again'. But thankfully, we reacted well and I thought it was a decent performance from us. There are still a few things to tinker with though and we should have won the game with more goals."
However Tilly continued that he was still more than pleased to be in the competition: "You look at the results in the F.A. Cup with Barrow beating Brentford on Friday night and Eastwood Town knocking out Wycombe and that goes to show what can happen in the Cup. The main thing for us was to make sure we got through to the third round. I thought Junior Stannislas did well on his debut and scored two good goals. He is an exciting player and it was a big plus for us to have him here today. I'm not sure I want to draw a smaller side at home to give us a bigger chance of reaching the fourth round or draw a bid side away. Of course, getting Manchester United away is the one everyone wants."
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