All three of the goals came in the final stages of the contest, with a brace for Burrows proving to be enough to secure the trophy for his side.
Peterborough have now won the competition for the second time, having previously triumphed in 2013-14.
Kwame Poku had the game's first half-chance in the 12th minute, with the midfielder finding space in an advanced area, but his effort was always rising over the Wycombe crossbar.
Wycombe had two opportunities of their own in the 18th minute, but Richard Kone had a strike blocked on the edge of the box, before Matt Butcher missed the crossbar with an effort moments later.
Garath McCleary had another opportunity for Wycombe in the 20th minute, but his volley from a dangerous area just missed the target.
Hector Kyprianou then had a chance down the other end, heading a Burrows cross over the goal in the 25th minute.
McCleary saw an effort kept out by Peterborough goalkeeper Jed Steer in the 30th minute, before Franco Ravizzoli did well down the other end to deal with a dangerous delivery from Jadel Katongo.
Peterborough had a shout for a penalty late in the first period, with a cross from Poku hitting Luke Leahy's arm, but the referee was not interested.
Wycombe goalkeeper Ravizzoli then did brilliantly to deny Katongo in the 41st minute, before Kone dragged one wide down the other end, with both sides threatening to make the breakthrough, but the half-time whistle was blown with the score at 0-0.
Kone came close to sending Wycombe ahead early in the second period with an inventive effort from a wide area, but the ball dropped over the crossbar.
Poku then fired over in the 55th minute, before Kieran Sadlier stuck too high in the 68th minute, with Wycombe on the front foot.
The deadlock was finally broken in the 85th minute, and it was Peterborough captain Burrows, who found the back of the net with a powerful strike off an assist from Ephron Mason-Clark.
Wycombe refused to be beaten from that effort, and the score was levelled in the 89th minute courtesy of a thumping Dale Taylor volley.
The drama was not finished, though, with Peterborough incredibly winning it in the first minute of added time at the end of the 90 through Burrows, who managed to lift a cross over the head of Ravizzoli from a wide position to secure the trophy for his team.