Two first-half goals to striker Aaron Huesser secured the win for Cambridge as the WaiBOP Championship moved into the second phase of the competition.
The 2-0 win, at John Kerkhof Park, means Cambridge leads the group of six clubs drawn to meet after the competition split into two.
Cambridge could not have wished for a better start as Huesser gave the home side the lead in the first minute of the match.
Despite injuries forcing first half replacements, Cambridge controlled the flow of the game, working well down the flanks with wide men Jordan Silvester and Joshua Dagnall stretching the Te Puke defence across the pitch.
This left space Cambridge to push through the middle, forcing Te Puke to give away free kicks in goalscoring positions.
Shaun Richardson hammered one free kick which skidded wide of the post and Cambridge forced a series of corners.
Huesser scored his second goal, meeting a 36th minute corner with his head and flicking it between two Te Puke defenders.
That turned out to be crucial goal and the sides played out a scoreless second half to give Cambridge three points.
Huesser was the Player of the Day.
Main photo: The ball edges over the Te Puke goal line as Aaron Huesser (obscured by post) prepares to celebrate his goal. More images below:
Cambridge 2 Te Puke United 0
Hamilton Wanderers 3 Tauranga Old Blues 0
Waikato Unicol 2 Tauranga City 11