AUSTIN, Texas — For a reserve point guard, Naadir Tharpe may be as confident a player as there is in America. It’s a quiet and understated self-belief, but it’s always there, the sort of confidence that leads to step-back three-pointers and jarring dribble-drives into the teeth of the defense.

It’s also the sort of coolness that led to this exchange between Tharpe and Kansas coach Bill Self with close to 8 minutes left Saturday in the Jayhawks’ 64-59 victory over Texas at the Frank Erwin Center. The fourth-ranked Jayhawks trailed a physical Texas team by six points — even after whittling down an 11-point deficit. And Tharpe was headed back to the bench.

He was scoreless at that point, and Self had jumped him at halftime for some passive play in the first half. But Tharpe didn’t want Self to forget about him.

“I asked Coach, ‘Can you get me back in there,” Tharpe said. “Give me another chance. And he said, ‘Don’t worry, I’m gonna get you back in there.’ ”

Two minutes later, Self made good on his promise, calling on Tharpe and employing a four-guard offense in the final 6 minutes. The small-ball unit helped Kansas tear off the restrictor plate, and the Jayhawks escaped by finishing the game on a 17-6 run over the last 6 minutes.

Texas had a chance to send the game into overtime in the final seconds, but guard Julien Lewis misfired on a three-pointer with 13 seconds left, and the Jayhawks corralled the rebound. It was another sweet escape for Kansas, which won its 15th straight and improved to 16-1 overall and 4-0 in the Big 12.

“It was tough,” said senior center Jeff Withey, who finished with 14 points and nine rebounds, “but we have a pretty mature team that continues to grind through.”

On Saturday, the formula included the four-guard scheme in the final minutes. With floor leader Elijah Johnson struggling all day — he finished one for 11 with three turnovers — Tharpe’s presence let Johnson slide off the ball and gave the Jayhawks another three-point threat. And within moments, the unit had produced a cannon-fire run that helped KU take a 54-53 lead on an alley-oop to freshman guard Ben McLemore. The sequence started on an up-and-under move from Withey in the post, and Tharpe followed by draining a crucial three-pointer.

“I think I was talking to the basketball at this point,” Tharpe said, “Like, ‘C’mon, you’re not gonna go in there for me again.’ The basketball gods definitely helped me in that one.”

The smaller, sleeker lineup also helped McLemore find some windows against a stifling Texas defense. McLemore, playing for the first time since suffering a minor ankle sprain against Baylor on Monday, finished the first half with just three shots. And sure enough, Self made sure to point that out during the intermission.

“I gotta do a better job of getting myself open and creating things for myself,” said McLemore, who finished with 16 points on six-of-10 shooting.

In the final minutes, McLemore finally got loose for a couple of dunks. And the chant of “Let’s Go Jayhawks” rang out in the Erwin Center, only to be drowned out by a chorus of boos. Still, the Longhorns, 8-9 overall and 0-4 in the Big 12, were rattled. And the Jayhawks were poised to strike.

“That’s the youngest team in the country out there playing,” Self said. “And they’re gonna be really, really good. … But we had some experienced guys playing down the stretch.”

But make no mistake: This was a slugfest, a matchup that had the rhythm of an amateur Lone Star square dance. (“We looked like a bunch of inexperienced kids off the playground,” Self said.) And with Kansas and Texas entering the day ranked first and second in the nation in field-goal percentage defense, the afternoon mostly played to form.

For Kansas, though, there’s still some concern in the offensive scuffles. KU has averaged just 61.6 points in its last three games, and a trip to rival Kansas State awaits on Tuesday night.

“We definitely are in a little funk,” Withey said.

But this Kansas team is mature enough to know that funks can come and go, but defense does not. And sometimes all you need is a late spurt — or a little guard off the bench — to get back in sync.

“We learned something today,” Johnson said. “Probably would have learned more if we lost, but we definitely learned something.”