After leading the opening laps of the race, Russell was overtaken by his team-mate for the lead.
However, as the rain started to fall the two Mercedes struggled in the slippery conditions and both ran off the track.
Their mistakes played into the hands of both McLarens, Lando Norris overtaking Hamilton for the lead.
Oscar Piastri also looked threatening, stealing second place off the seven-time world champion to make it a McLaren 1-2 at the front.
Russell had little chance to fight back and was forced to come into the pits soon after his pit stop.
"Probably about 10 laps before I had to retire. I saw temperature alarms coming up my steering wheel and some sort of water cooling failure," Russell said to Sky Sports after the race.
"I don't really know what to say. The car was feeling so good.
"I knew it was going to be a long race. I struggled a bit around the damp patches but very gutting."