New Zealand won by three wickets after being tested in both batting and bowling with Guptill making his first hundred of the tournament and Southee hitting the winning six and the boundary to seal the win.
Bangladesh were lifted to 288-7 by Mohammad Mahmudullah's 128 not out -- his second successive hundred in the tournament.
"It was a bit of a topsy-turvy game. They were on top for a lot of it, then we fought back and then they fought back again. It's just nice to get the win under the belt and show some fight towards the end, and the way Timmy and Dan (Vettori) finished it off was pretty fantastic," said Guptill.
Southee, who finished wicketless, came in to bat with 20 still needed off 16 balls.
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"Obviously a tight finish, and it just shows we found another way to win another game of cricket, and I think that's a testament to the side," said Southee.
"We were challenged in different ways, and we managed to come through it and add another two points to the total. Yeah, it's a different win for us, but it just shows us the quality in the side and we can win when we're in difficult circumstances throughout the game."