Guptill's World Cup record 237 not out propelled New Zealand to 393 for six in their quarter-final against the West Indies who fell apart in the face of Boult's four for 44 and were all out for 250.
The 143-run victory put New Zealand into the semi-finals for the seventh time where they will meet South Africa with neither side having ever made the final.
Guptill said he was proud of what he achieved "and hopefully we can win another game and then another one after that."
"We knew they (West Indies) have had their troubles but we knew they would be competitive. They've virtually got 11 match winners in their team," Boult said, overlooking New Zealand's thumping victory with more than 19 overs to spare.
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"Now we can focus on the South Africans and they're going to be as equal a challenge if not better.
"We will scout them and prepare for that game like we have for every other one, and set the game plan down and give it our best on Tuesday."
Samuels was in full flight with 27 off 16 deliveries when he attempted to pull Boult over the boundary only to see the 36-year-old Vettori leap into the air and take a remarkable one-handed catch.
"I couldn't believe it. Leaping up a couple of metres in the air and sticking out his paw.
"I actually enjoyed the celebration afterwards which was a bit of strut. The best sight was seeing everyone sprint towards him it was something pretty cool," added Boult, who is now the leading wicket-taker at the World Cup with 19.