Southee took 4-19 off 16.2 overs on the third day of the first Test in Kingston, Jamaica, as the visitors bowled the West Indies out for 262 to claim a 246-run first innings lead.
One of the 25-year-old’s scalps was Gayle (64), with the explosive opening batsman edging to New Zealand wicketkeeper BJ Watling.
Gayle conceded he could have been out even earlier with Southee bowling so well at Sabina Park.
“From my personal point of view, I thought Southee bowled well all morning,” the 34-year-old said.
“I survived that first session, which was a vital one. I guess he deserved a wicket in his second spell, the way he bowled in the morning.”
Southee and New Zealand’s debutant off-spinner Mark Craig (4-91 off 24 overs) destroyed their hosts on day three but two late wickets for the West Indies saw the tourists end the day’s play at 14-2.
Gayle claimed the home side could still save the Test and he pledged to shoulder the run-scoring responsibility in the West Indies’ second innings.
“I am looking forward to batting second and hopefully I can get a big one,” he said.
Despite the cheap dismissals of Peter Fulton and Kane Williamson just before stumps, New Zealand were buoyant following the third day with a rare Test victory in the Caribbean in their sights.
New Zealand have only defeated the West Indies at home once in their Test history – back in 2002.
If they win the first Test at Sabina Park, Craig will have played a key role on debut and the 27-year-old was thrilled with the way he bowled on Tuesday.
Craig claimed two wickets in three deliveries in the 19th over of the day, trapping Kieran Powell (28) lbw before Kirk Edwards edged the ball to Ross Taylor at first slip.
“It was nice to go bang-bang there, bit of a dream start,” Craig said.
Craig also claimed the wickets of Kemar Roach and Sulieman Benn but was quick to praise Southee as the main man with the ball.
“It was very nice to be able to contribute to the team today and obviously to get a couple of wickets was a bonus,” Craig said.
“Timmy bowled exceptionally well, set the standard for us up front, it was really nice to chip in with a couple myself.”
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