Australian Webber, who claimed the ninth victory of his Formula One career on Sunday by taking the chequered flag in the British Grand Prix at Silverstone, had been linked with a move to the Italian team next year.
But the 35-year-old, firmly in the title hunt this season as he trails Ferrari's Fernando Alonso by 13 points in the standings, has opted to remain with Red Bull for what will now be a seventh season.
Webber, who hinted over the weekend he was close to tying up his future, said: "I've been with Red Bull Racing since 2007 and have achieved nine grand prix wins during that time.
"I'm high on confidence at the moment and firing on all cylinders. "I know the team well and I'm very comfortable here. We have grown together over the years and it feels like absolutely the right thing to stay with Red Bull for another season.
"The team is constantly working hard to improve in all areas and we've shown that together we can win races.
"It's great to be able to make this announcement off the back of the win in Silverstone at the weekend and I'm looking forward to competing on the edge and pushing myself in every race again next season."
He admitted, however, that talks had taken place between himself and Ferrari over a move which would have seen him link up with Fernando Alonso.
There were discussions with Ferrari, but my decision was to stay here," he said.
"With Red Bull I virtually know everyone's shoe size!
"I know everyone's names, I know how everyone ticks and I know all the departments, which is very, very important for a driver. There's a whole raft of factors.
"So in the end, it was an easy decision. It's been an interesting few months, but overall the continuity and desire for me to continue at Red Bull Racing was very strong."
Horner, who last week said Webber's motivation and driving this season could not be faulted, has moved quickly this year to secure his services.
Horner said: "Mark has driven very well in the first nine races of this season and his performance has been impressive.
"Much of his Formula One success has been during his time with Red Bull Racing and together we have achieved 10 poles, nine wins and 31 podiums.
"As there was a strong desire from both sides to continue the partnership, it was a logical decision to extend our relationship and it is with great pleasure we confirm Mark will drive for us in 2013."
With team-mate Sebastian Vettel already contracted to Red Bull Racing until the end of 2014, the Milton Keynes-based marque are the first team to confirm their 2013 driver line-up.
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