But the Huskies took control in the second half and pulled away for another win.
Kaleena Mosqueda-Lewis had 19 points to lead four Connecticut players in double figures, and the defending national champion Huskies shook off BYU early in the second half to win 70-51 in the NCAA women's regional semifinals Saturday.
''There's always one of these in the tournament,'' UConn coach Geno Auriemma said. ''I hope there's only one.''
The Huskies (37-0), winners of 43 straight, need a win over Texas A&M or DePaul on Monday night to reach the Final Four for the seventh straight year.
UConn season scoring leaders Breanna Stewart and Bria Hartley overcame slow starts, with Stewart having 12 of her 16 in the second half and Hartley all 12 after halftime.
Hartley scored 11 of the Huskies' 13 points during a stretch their lead grew to 50-41.
''To be honest with you, people on our staff will vouch for me, I knew it was going to be like this,'' Auriemma said. ''I knew it wasn't going to be an easy matchup. It's not supposed to be easy, no matter how we make it look sometimes. I think we showed a lot of who we are in the second half. When we need a play, we made the plays.''
Kim Beeston led the Cougars (28-7) with 16 points, and Morgan Bailey added 14. Jennifer Hamson had nine points to go with 13 rebounds and six blocked shots.
Mosqueda-Lewis and Stefanie Dolson each grabbed 13 rebounds, with Mosqueda-Lewis gathering eight on the offensive end and scoring on putbacks.
''All of us take it upon ourselves, if a couple people aren't hitting shots like they usually do, it's our job to pick them up and make some shots,'' Mosqueda-Lewis said. ''I think especially with our starting five, we have accountability.''
The Cougars, who came to Lincoln off wins in 12 of 13 games, joined Kansas in 2013 and San Francisco in 1996 as the only No. 12 seeds to make it to the regional semifinals.
They were trying to become the lowest seed to reach the Elite Eight since No. 11 Gonzaga did it in 2011.
Until early in the second half it looked as if they had a chance to do it - against the dominant program of this era, no less.
''We worked hard to get here,'' Hamson said. ''I'm proud of my team and what we've done together. Hopefully, they can take this and move forward.''
UConn won the only previous meetings of the programs, in 2007 and 2008, by 46 and 49 points.
BYU showed no sign of being intimidated against the Huskies' cast of stars. The proof? UConn was down 35-34 with 17:34 left - the latest it has trailed this season - after Lexi Eaton hit three straight baskets to open the second half.
Hartley's 3-pointer, the first after nine straight misses from distance by UConn, started a 13-2 run that put the Huskies in control. Mosqueda-Lewis made a 3-pointer, then stole the ball on the other end and threw a baseball pass that Hartley scored on.
The Huskies kept their transition game going when Moriah Jefferson delivered a long pass to Hartley for another layup that stretched it to 47-37.
''The key to slow them down is to make baskets and not let them fast break,'' BYU coach Jeff Judkins said. ''They run like deer and they jump out of the gym.''
Beeston scored 13 points with three high-arching 3-pointers in the first half as the Cougars put the Pinnacle Bank Arena crowd on upset alert, and she hit a fourth midway through the second half to make it a six-point game.
That's as close as the Cougars would come to doing the improbable.
''We were just excited knowing we were playing great basketball with the best of the best,'' Beeston said. ''We knew if things kept going the way we wanted to it would be a close game down to the wire. We're a little disappointed we couldn't do that.''
UConn trailed for the first time in the tournament at 3-2 when Bailey made the first of her three 3-pointers just over a minute after tipoff. The Cougars led by as many as six at 27-21 on a Beeston jumper before UConn came back late in the half.
''They're bigger and stronger, and they ended up wearing us down, I think,'' Judkins said. ''Their best player didn't have her best game tonight, but they're so balanced.''