The home side had the better chances of the opening half hour and Stags loanee David Grof pulled off a fantastic save in the 11th minute. George Pilkington's header seemed to have him beaten but his one-handed leap clawed the ball to safety.
Owusu saw his turn and shot rebound off the inside of the post but that scare seemed to spark Mansfield into life.
Danny Mitchley should have done better but his weak lob fell gratefully into the arms of former Stags goalkeeper Pilkington, who had a loan spell with his former club earlier in the season.
Paul Connor fired wide from close range after a well-worked corner and Mitchely had another chance from 18 yards out but failed to hit the target.
Luton had the best possible start within a minute of the interval, Owusu's shot from the edge of the box flew into the top left-hand corner following a soft header away from Kevin Sandwith.
Oswusu then had a great chance to put Luton ahead in the tie, as he rounded Grof but fired over the bar from eight yards out with two Mansfield players on the line.
Luton lost their cool as normal time came to an end, Gnakpa was sent off for a second yellow card in the 84th minute and Lawless was shown a straight red for an elbow on Kyle Nix.
With the tie level on aggregate at 1-1 after 90 minutes, Mansfield dominated the first half of extra time with no end product as they were unable to test Pilkington.
But with the game looking set to go to a penalty shoot-out, Mansfield were awarded a penalty just two minutes from time as Briscoe's shot hit an arm in the box.
The winger stepped up to take the penalty himself and, despite Pilkington saving his shot straight down the middle, Briscoe fired in the rebound to send Mansfield to Wembley for the first time in 24 years.
Luton: Kevin Pilkington, Graham, George Pilkington, Kroca, Howells, Newton, Keane, Lawless, Gnakpa, Barnes-Homer, Owusu. Subs Not Used: Morgan-Smith, Crow, Walker, Poku, Tyler.
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