Francis sprinted down the backstretch in the 1,600-meter relay to catch Texas' Ashley Spencer at the tape, giving the Ducks 44 points—a half-point better than the Longhorns for a fifth straight championship.
Francis and teammates Chizoba Okodogbe, Laura Roesler, Christian Brennan finished in 3 minutes, 27.40 seconds, just ahead of Texas at 3:27.42, for a college indoor mark. Earlier, Francis set the indoor record in the 400 at 50.46 seconds. She was even faster in the relay, finishing her leg in 50.44.
Oregon became the first school to sweep the team titles since Arizona State in 2008.
"Never in a million years did I think that," Oregon coach Robert Johnson said.
The Ducks won the men's title with 62 points, eight more than Arkansas.
"Overall a great team performance for the men," Johnson said. "We thought that we were a little bit under-ranked. We thought we were better than the scoring tables and all the dope sheets had us at. We just knew that if we got to the meet healthy, we had a great shot."
The key, Johnson said, was Cheserek's performance. He won the 5,000 on Friday night in 13:46.67, and took the 3,000 on Saturday in 8:11.59.
"Outstanding performances by Edward Cheserek to get two national championships there," Johnson said.
For the women, it came down to the final leg of the final race in the closest of decisions.
"I thought. 'Oh my gosh, did we really leave it that late,'" Johnson said. "But she got them at the end."
Francis said her teammates' performance on the opening three legs helped spur her on.
"Seeing my teammates' effort, they were really pushing," she said. "I didn't want to let them down. I totally went for it. I saw my team cheering us on and we just pushed us to the next limit. We won, the men's team won, the woman's team. It's just mind blowing right now."
Spencer said she knew Francis was right there and did everything she could to hold her off.
"I definitely felt her coming and I just tried to give it my all over the last 50," Spencer said. "I really pushed it, I pushed it. I tried and I tried. And she got the best of me. Phyllis Francis is an amazing athlete. I knew it wasn't going to be easy. But it was a great race."
Dartmouth's Abbey D'Agostino continued to add to her stash of championships, winning the 3,000 in 9:14.47 after taking the 5,000 on Friday, giving her six individual titles.
In the mile, Anthony Rotich Jr. of Texas El Paso won in 4:02.54, and Emily Lipari of Villanova took the women's race in 4:38.82.
In one of the closest events of the day, Omar McLeod of Arkansas won the 60 hurdles in 7.586, just ahead of Florida's Eddie Lovett in 7.588. It was the same for the women, with Sharika Nelvis of Arizona State finishing in 7.923, just ahead of Baylor's Tiffani McReynolds in 7.930.
Florida State's Dentarius Locke won the 60 in 6.52, and Remona Burchell of Alabama won the women's race in 7.11. Deon Lendore of Texas A&M won the 400 run in 45.21, Mississippi State's Brandon McBride took the 800 in 1:48.17, and Roesler helped the Ducks' cause with a women's win in 2:03.85.
Louisiana State won the 1,600 relay in 3:04.10.
In field events, Ryan Crouser of Texas took the shot put at 69 feet, 7 inches, and Christina Hillman of Iowa State won the women's title in 59-6 3/4.
Baylor's Felix Obi went 54-5 1/4 to win the long jump, and Shanieka Thomas of San Diego State took the women's event in 45-10. Akron's Shawn Barber won the pole vault at 18-10 1/4.
Local star Curtis Beach of Duke came home to win the heptathlon with 6,190 points, and Kendell Williams of Georgia finished with 4,635 points for the women's title.