In what manner was the Vympel R-73 (AA-11 Archer) a game-changer in comparison with NATO's missiles?
The Russians brought some new ideas to the missile warfare when they mated the Vympel R-73 with Mig-29. The most important of these was the high offset sensor lock mode, which allowed the pilot to 'lock on' to the targer regardless of where the nose was pointed at. According to Peter “Stoini” Steiniger a JG 73 Pilot, who flew Mig-29s in East Germany,

The nice airframe in combination with one weapon was the killer: The Archer in [sensor lock] mode.” Introduced in the mid-1980s, the Archer AA-11 is a very capable heat-seeker with a greater range than the U.S. Sidewinder. “A simple monocular lens in front of my right eye enabled me to slew the seeker head of the missile onto my adversary at high angle off [target].” The Fulcrum’s ability to lock a missile even though its nose was pointed far away from its target “watered many eyes,” ...

The image below shows the missile aiming sight mounted on the helmet of a JG 73 pilot.

Image from airspacemag.com
This off boresignt advantage was not achieved by the western forces till they introduced the AIM-9X missile and the Joint Helmet-mounted Cueing System in 2002.
Aircraft-wise, the Mig-29 was a revelation to the west as the previous Russian designs have not been so sophisticated, so as to speak. The West was concerned enough with Mig-29 to start looking for new fighter ideas. Another important equipment in Mig-29 was the Infra Red Search and Track System (IRSTS), an idea for which the US is just warming up. Most of the exchange pilots, who flew the Mig-29 against the US F-15s and 16s were quite surprised and initial engagements usually went the 29s way, though techniques were later devloped exploting the Mig-29s' weaknesses. From an USAF exchange pilot:

From BVR (beyond visual range), the MiG-29 is totally outclassed by western fighters. Lack of situation awareness and the short range of the AA-10A missile compared to the AMRAAM means the NATO fighter is going to have to be having a really bad day for the Fulcrum pilot to be successful.
In the WVR (within visual range) arena, a skilled MiG-29 pilot can give an [F-15] Eagle or [F-16] Viper driver all he/she wants.