There exists no precise explanation of Yuri's almost preternatural ability to follow Henry Brogan's aventure.
As you mention, the tracking chip wouldn't have helped Yuri to watch Henry's escape. This means he either had a direct access to the DEA's strike team's own cameras or watched Henry Brogan's house through different means. The latter solution somehow feels more unlikely has Henry as been depicted as particularly cautious with security, and his domain's numerous alarms and security systems would probably have detected a spy installing hidden cameras around the property.
The next paragraph is only a guess : Yuri had a personal interest in Brogan from the get-go, as he willingly admits. He obtained official funds and resources to spy on Brogan both when the russian scientist (Dormov) defected to Gemini to work on the clone program, and when the russian secret services decided to use Henry Brogan as a proxy to annihilate Gemini's operations. Unless Yuri has secret contacts within the DEA, the SVR (a KGB-like real-world russian organization) most probably hacked the DEA at some point to follow the strike team using those funds and resources. Or something similar.
We know the russian secret services are intending to use Henry Brogan as a proxy thanks to the following line (present both in the movie and within the official novelization) :
"That is what we are doing [taking out Gemini's facility] -- except you are the missile ! Good luck !"
As mentioned by another comment, this off-hand line from Yuri was probably meant to evoke governments and fictionalized secret services' uncanny ability to spy on targets with quite the plot-convenient efficiency. The scriptwriters / director probably didn't dwell much on the actual implications, since Yuri and the russians drop out of the movie at that point.