Russell, who joined Derby from Dundee United this summer, took just nine minutes to open the scoring at Dalymount Park in Dublin, sweeping home Michael Jacobs' cut-back.
The home side levelled five minutes later through Chris Lyons, only for Russell to head home his second goal in the 26th minute.
County manager Nigel Clough fielded a different side in the second half and goals from Callum Ball, Ben Davies, Paul Coutts and Conor Sammon sealed a convincing victory.
Defender Richard Keogh hailed the result as a great start to Derby's pre-season campaign.
He told the club's official website: "As much as pre-season is about fitness, it was great to be able to blow the cobwebs away today and for each player to get 45 minutes under their belt.
"It is nice to get back into a match environment after two weeks of hard work - which has gone really well.
"It was a good game for us to start our pre-season with and to score six goals tops everything off."