Tennessee basketball proved it could win at Rupp Arena in the first two rounds of the NCAA tournament. Next, it might have to prove it can beat Rupp Arena’s home team.

That might present a greater challenge than what the Vols encountered in an impressive 67-58 second-round victory over UCLA on Saturday in Lexington, Kentucky.

Or maybe, the Vols will avoid tangling with their longtime SEC rival. Third-seeded Kentucky, which beat UT twice during the regular season, will have to get past Illinois in a second-round game Sunday to create a Tennessee-Kentucky matchup in the Sweet 16.

Not only did No. 2 seed Tennessee (29-7) advance to the tournament’s second weekend with the victory over No. 7 seed UCLA (23-11). It moved within one game of a second 30-win season under coach Rick Barnes.

The game between a pair of defensive-minded teams unfolded as expected in the early going. The Vols finally managed separation when they ran off eight consecutive points to close out the first half with a 32-25 lead. A Chaz Lanier steal, which he converted into a three-point play, highlighted UT’s surge.

As good as UCLA’s defense has been at times this season, it struggled terribly keeping up with Lanier. He reached the 20-point mark early in the second half. By then, he also had broken Chris Lofton’s school record for 3-pointers.

Lanier, who scored 29 points in the first-round victory over Wofford, has been at his best in the tournament, an encouraging sign for the Vols, whose March Madness failures under Barnes usually have been related to offensive shortcomings.

They had more offense than coach Mick Cronin’s Bruins could handle – and more defense, too.

Tennessee joined two other top-four Midwest Regional seeds in reaching the Sweet 16. No. 1 seed Houston started fast against Gonzaga and held on for an 81-76 victory. Fourth-seeded Purdue cruised past McNeese 76-62 into the Sweet 16. No 3 Kentucky can make it four-for-four by turning back No. 6 Illinois on Sunday.

The Vols are assured of a familiar opponent in the Sweet 16 regardless of whether Illinois or Kentucky advance.

UT edged Illinois 66-64 in Champaign in mid-December, thanks mainly to the scoring of guard Jordan Gainey, who had a game-high 23 points, including the game-winning basket in the final seconds.

Kentucky is a different matter. The Wildcats defeated the Vols twice during the regular season, which could evoke different feelings about a third meeting among the Vols fanbase.

Tennessee optimists might cling to the basketball adage: “It’s hard to beat a team three times in a season.” But if you watched both Kentucky-Tennessee regular-season games, you can’t easily dismiss the Wildcats as a possibly serious obstacle to the Vols reaching the Elite Eight.

Kentucky didn’t just win twice. It won as so few teams have against the Vols, exploiting their usually dogged defense with surprisingly efficient shooting.

In the first Tennessee game, Kentucky made 50% of its shots in a 78-73 victory at Food City Center. The Wildcats were just as accurate in winning the rematch at Rupp Arena 75-64. In each game, they made 12 of 24 3-pointers.

Those outcomes are two reasons why UT fans might ignore the “hard to beat a team three times” saying and pull for Illinois, which is a 1.5-point favorite over Kentucky despite being a lower seed.

ADAMS: Tennessee basketball did enough to convince me it’s headed for a Houston showdown

Tennessee fans shouldn’t complain if Illinois is victorious. But looking further ahead, they probably feel better about their chances against Houston if the region’s top two seeds wind up meeting in the Sweet 16. Houston looked vulnerable in the second half against Gonzaga, which cut the Cougars’ double-figure lead to a point with 21 seconds remaining.

Unlike Houston, the Vols raised their level of play as the game progressed against UCLA. That performance should bolster their confidence for the Sweet 16 when they try to become the first team in school history to make the Final Four.

 John Adams is a senior columnist. He may be reached at 865-342-6284 or john.adams@knoxnews.com. Follow him at: twitter.com/johnadamskns