The Italian company will introduce modified tyres for this weekend's German Grand Prix in an attempt to ensure the blow-outs are not repeated.
But it has requested that the sport's governing body, the FIA, steps in to ensure teams run the tyres correctly.
A new range of tyres will be developed for the Hungarian GP later this month.
The Silverstone six
Left rear explodes on lap eight
Right rear deflates on lap nine
Left rear blows on lap 10
Left rear blows on lap 15
Front left goes on lap 29
Left rear blows on lap 47
A statement from Pirelli insisted the 2013 tyre "does not compromise driver safety in any way if used in the correct manner".
Pirelli attributed blame to the teams mounting the rear tyres the wrong way around, running low tyre pressures, using extreme cambers - the angle at which the wheel is mounted on the suspension - and what it described as "high kerbs" at Silverstone.
In a second statement released 90 minutes after the first, Pirelli motorsport director Paul Hembery said: "In no way are we intending to create arguments or attack anybody. We have taken our responsibilities upon ourselves.
"But not having full control over all the elements that impact on the use of the tyres, we need everybody's contribution. With regard to this, we are receiving the full support of all the parties involved, for which we are very grateful."
The manufacturer said, in its first statement, that all the failures happened on cars with the rear tyres mounted on opposite sides of the car from the one intended.
This has become common practice this season as teams try to manage usage of tyres that have been designed to wear out quickly to guarantee races with multiple pit stops.
Pirelli was aware of the practice but has now admitted it "underestimated" the effect it could have and admitted it "did not forbid" it.
It added that "under-inflation of the tyres and extreme camber settings, over which Pirelli has no control, are choices that can be dangerous under certain circumstances.
"Because of this, Pirelli has asked the FIA for these parameters which will be a topic of accurate and future examinations.
"Pirelli has also asked for compliance with these rules to be checked by a dedicated (FIA) delegate."
The company added that it wanted the usage of tyres "to be regulated and carefully controlled by Pirelli itself."
This is not allowed under current rules but the company has asked the FIA to change the regulations to ensure that could happen.
Despite saying the tyres would have been safe if operated in the correct manner, Pirelli will introduce a new construction for this weekend's race at the Nurburgring.
These will be "easier to manage", according to Hembery.
The tyres will use a belt - the material on which the rubber is mounted - made of Kevlar rather than the current steel. Kevlar is designed to be more puncture resistant and it will also make the tyres run about 10C cooler than before.
Pirelli had wanted to introduce these tyres for the British Grand Prix to reduce the risk of the delaminations - when the tread strips off the tyre - seen at races earlier this season.
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But wet weather meant the teams were unable to do enough laps to reassure them the tyres would not change the behaviour of the cars and potentially favour one team over another.
From the Hungarian GP on 26-28 July onwards, Pirelli will introduce a new range of tyres with the same construction as was used last year with the rubber compounds used in 2013.
These tyres also have a Kevlar belt but the construction is different from those used this year.
This will almost certainly change the way the cars behave and could favour some teams over others.
Hembery said: "What happened at Silverstone was completely unexpected and it was the first time that anything like this has ever occurred in more than a century of Pirelli in motorsport.
"These incidents, which have upset us greatly, have stressed the urgency of the changes that we already suggested - which will be introduced during free practice in Germany on Friday.
"We would like to acknowledge the willingness of the FIA, (F1's commercial arm) FOM, the teams and drivers to act quickly to find an immediate solution to the problem.
"In particular, the adoption of winter tests, arranged with the FIA, that are more suitable for tyre development and the possibility of carrying out in-season testing will contribute to the realisation of tyres with increasingly improved standards of safety and performance."
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