As much as I'm a Typhoon fan, and I believe that in a dogfight it'd beat any F-35 variant, I can't see it beating a Raptor in most situations. The F-22 will almost certainly see the Typhoon first with its own AESA LPI radar (APG-77V1). No Typhoons yet have AESA, and although the Captor is great for a mechanical radar, it is now outdated.I do believe it's LPI, but it is likely less LPI than the APG-77V1. The IRST (PIRATE) will probably be the only hope of detection that the Typhoon has, and it's worth noting that the Raptor does reportedly have a reduced heat signature, as well as its low RCS. Clouds and other atmospheric conditions can reduce with range further, as will the fact that the F-22 will be head-on with the Typhoon, and due to supercruise, doesn't have to use afterburners.
Bottom line is that the Typhoon probably won't detect the F-22 prior to its missiles being launched. Maybe it'll get a hit off the Raptor's weapons bay opening.
It's also worth noting that the Raptor can carry x6 AIM-120s internally (plus x2 AIM-9Xs). IIRC, an exercise between the UK, US, and French (Atlantic Trident?) apparently demonstrated that 2-3 AAMs are required to destroy a modern fast jet in a contested environment with ECM, decoys, etc.
The only time I can see a Eurofighter having an advantage is when the F-22 is in a IRM only battle - as you stated, the ASRAAM does out range even the latest AIM-9X; plus the AIM-132A has been proven to destroy a target 180 degrees behind the firing fighter. Worst case scenario here should be a mutual kill.
In a guns only dogfight, the Raptor probably has a slight advantage due to thrust vectoring, but it'll certainly come down to pilot skill over anything.