Red Bull have led the championship since 2022 and enjoyed the most dominant season on record last year with 21 wins in 22 races.
Oscar Piastri's Azerbaijan Grand Prix win and Lando Norris's fourth place sent McLaren top of the standings, 20 points clear of Red Bull who had Sergio Perez crash out while chasing the podium and Max Verstappen finish fifth.
Triple world champion Verstappen's lead over closest rival Norris was trimmed to 59 points from a previous 62 with seven rounds remaining.
Horner said the result was a "big hit" but there was still everything to play for.
"Look, we're now not defending, we're chasing. So it changes the dynamic again, and we're just going to throw everything at it," he told reporters.
"We've got 20 points as a deficit now, so we've got to attack."
Red Bull had looked in much better shape going into the race than at some recent ones, with Perez and Verstappen starting fourth and sixth respectively while Piastri lined up second and Norris 15th.
A crash between Perez and Ferrari's Carlos Sainz on the penultimate lap, as well as a strong drive by Norris and good McLaren teamwork, turned everything around and former champions McLaren took over at the top for the first time since 2014.
"It's frustrating, particularly after where Lando qualified, that we didn't beat him today but thankfully he hasn't scored big points," said Horner.
Perez, who has failed to finish in the top three in the last 12 rounds, told reporters he was "super sad" for the team but recognised positives in terms of car performance.
"It's a massive frustration," he said after the race. "But the positive thing is that the pace is back."
