Swift broke Johnny Rodgers' record on the first of his two touchdowns. Swift, who had 121 yards in receptions, caught a career-high 11 passes to increase his four-year total to 147. That's four more than Rodgers, the 1972 Heisman Trophy winner who set the record in three years.
Joe Ganz's 9-yard pass to Swift late in the third quarter erased Baylor's 20-17 halftime lead, and the two teamed up for a 53-yarder on the Huskers' next series to make it 30-20.
Baylor (3-5, 1-3) came in 2-47 in Big 12 road games, including losses in eight straight. But the Bears gave Nebraska (5-3, 2-2) a scare, with Griffin always a threat to break a long run. He finished with 121 yards rushing, and he threw for 134.
The Huskers started their go-ahead 80-yard drive after Ben Parks missed a 19-yard field-goal attempt. Ganz scrambled and completed a 15-yard pass to Swift on third-and-10 before hitting him with the record-breaking touchdown pass on a short square-out on the left side of the end zone.
Baylor had its ensuing possession kept alive by a roughing-the-punter penalty, but Nebraska got the ball back when safety Larry Asante sacked Griffin on fourth-and-4 at the Nebraska 31.
Ganz, 32-of-46 for 336 yards and three touchdowns, found Swift on a deep fly pattern to make it a 10-point game.
Finley finished with 88 yards for Baylor and Marlon Lucky with 83 for Nebraska.
Baylor gave Nebraska its most competitive game in Lincoln since the Bears won a non-conference game 26-7 in 1956.
Four of Nebraska's previous six home wins over Baylor had been by shutout. In six meetings as Big 12 opponents, Nebraska had outscored the Bears 277-69, scoring 48 or more points five times.
All that was ancient history Saturday.
Baylor scored two quick touchdowns after Nebraska opened a 7-0 lead on Ganz's 6-yard pass to Ryan Hill.
The Bears picked up 15 yards on a personal foul by Cody Glenn for making a horse-collar tackle on Griffin, and then Finley busted through the line for a 32-yard touchdown.
Griffin broke two long runs on the Bears' next series, starting with a 40-yarder on the first play and a 47-yarder around left end for the tiebreaking touchdown on fourth-and-1.
Nebraska was up 17-14 after Lucky's 18-yard run, but Griffin led the Bears 80 yards in six plays just before half. After a facemask penalty on Glenn, Griffin passed 35 and 16 yards to Ernest Smith before Jacoby Jones scored from the 1 for a 20-17 halftime lead.