Young was intercepted twice by Jessie Bates in a rough start for the former Alabama star who is being counted on to lead a resurgence in Carolina.
Robinson, selected at No. 8 by the Falcons in defiance of those who say running backs shouldn't go that high, showed his versatility by taking a swing pass for an 11-yard touchdown.
Allgeier added a clinching touchdown on another 3-yard run with just under five minutes remaining.
It was a rough day for both offenses, each of them led by a young quarterback with plenty to prove.
Desmond Ridder, who started the final four games of his rookie season to set up his first full year at the helm in Atlanta, couldn't get much of anything going in the deep passing game. He was sacked four times and finished 15 of 18 for just 116 yards.
It was an even tougher day for Young, whose Heisman Trophy-winning career at Alabama led the Panthers to trade up for the No. 1 pick and select him as the first of three quarterbacks who went in the first four spots of the draft.
But Young looked like a rookie on two nearly identical throws across the middle, both times failing to pick up Bates lurking in the vicinity. The rookie also had a fumbled exchange, took a delay of game penalty and was sacked twice.
He finished 20 of 38 for 146 yards and failed to become the first quarterback since David Carr in 2002 to win his starting debut after being selected at the top of the draft. The last 15 quarterbacks with that honor are now 0-14-1 in their first starts.
Robinson's first NFL TD was one to remember, set up by Bates' first interception that gave Atlanta the ball at the Panthers 17.
Taking a short pass from Ridder, the rookie faked out linebacker Frankie Luvu, smashed through two would-be tacklers and burst into the end zone.
After sitting out a couple of practices, casting doubt on whether he would play without a new contract, Carolina outside linebacker Brian Burns made a strong pitch for a long-term deal.
The two-time Pro Bowler seemed to spend about as much time in the Falcons backfield as Ridder. Burns had 1 1/2 sacks, another hit on the quarterback, and planted Robinson for a big loss.
Panthers: CB Jaycee Horn went out in the first half with a hamstring injury and did not return. ... WR D.J. Chark Jr. (hamstring) was ruled out before the game.
Panthers: Take on another NFC South rival when the New Orleans Saints visit Charlotte on Sept. 18 for Carolina's home opener on Monday night football.
Falcons: Play their second straight home game to begin the season when the Green Bay Packers visit Atlanta next Sunday.
AP NFL: https://apnews.com/hub/nfl
Paul Newberry, The Associated Press