Along with the rookie quarterback's big day, Kmet chipped in by handling emergency long-snapping duties as the Bears (4-2) won their third straight game for the first time in nearly four years.
Allen led a tea party celebration after his 9-yard touchdown reception in the third quarter and his second score gave Chicago a 28-10 early in the fourth. D'Andre Swift added a 1-yard plunge for the Bears.
The Jags (1-5) started fast but Gabe Davis dropped a pass in the end zone — after he was whistled for a false start — and Jacksonville settled for a field goal on its opening drive.
Davis later caught two second-half touchdown passes from Trevor Lawrence — the second made the score 35-16.
Kmet broke two tackles on a 31-yard touchdown reception to give Chicago a 7-3 lead in the second quarter when the Bears mixed in no-huddle to spark their offense. He then snapped for the extra point because long snapper Scott Daly left the game earlier with a knee injury.
Williams found Kmet in the end zone with 13 seconds left in the half to complete an 85-yard drive than included the rookie quarterback scrambling for 23 yards on one play and 19 on another.
Williams, the No. 1 pick in this year's NFL draft, finished 23 of 29 for 226 yards with the four TD passes and one interception. He picked up 56 yards on four carries.
Kmet had five receptions for 70 yards. Swift ran 17 times for 91 yards.
Jags coach Doug Pederson, who got a vote of confidence from ownership on Saturday, told the NFL Network at halftime while trailing 14-3 that his team had "self-inflicted wounds."
Jags tight end Evan Engram, in his first action since Week 1, fumbled after a reception on Jacksonville's first drive of the third quarter. The Bears turned that into points when Allen scored his first touchdown with the team.
Lawrence completed 23 of 35 passes for 234 yards, two touchdown passes and one interception. Engram had 10 catches for 102 yards.
It's Chicago's first three-game winning streak since late in the 2020 season.
Louis Rees-Zammit, the Welsh rugby star who is on Jacksonville's practice squad, made the trip and chatted with NFL Commissioner Roger Goodell on the field before the game. Goodell has led the league's push for international growth.