Spurs won 3-1 with Will Lankshear scoring twice and George Abbott also on the scoersheet. Timur Tuterov scored a late consolation for Sunderland.
Burnett told SPURSPLAY: I'm so happy for everyone involved for the players, for the staff, for the Club it's a really good achievement for us this season.
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"We're feeling a lot of happiness right now. I'm so proud of the players, they've given everything this season, we finished top of the table and went into a new format with the knockout stage, but we deserved to be in the final.
"We've had some really tough games but we've come through them and I'm just so pleased that we've managed to have one final push and do what we set out to do."
Lankshear was also named Premier League 2 Player of the Season.