Evans had a weekly reach of 9.23m listeners in the first three months of 2012, while Moyles could only manage 7.1m, according to the latest Rajar data.
This marks Evans's biggest lead over Moyles since he held a 1.7m advantage in the third quarter of 2011.
It also helped Radio 2 post a weekly reach of 14.56m, the station's second highest ever total and up strongly from 14.27m in the previous three months. Radio 2's audience share of 16.8% was up from 16% at the same period last year.
Alongside Evans, Radio 2 presenters Ken Bruce and Jeremy Vine also racked up record audiences, at 7.72 million and 6.76 million respectively.
In contrast, Radio 1 saw its weekly audience fall to 11.14m, down from 11.67m last quarter and 11.83m last year. The station's audience share slumped to 8.2%, from 8.5% last quarter and 8.7% last year.
Over the three months, Moyles saw his breakfast audience fall from 7.24m in the final three months of 2011, and 7.52m last year.
Fresh from being named UK Station of the Year at the Sony Radio Academy Awards, digital station Radio 6 Music posted record listening figures in the first quarter.
The home of DJs such as Lauren Laverne, Jarvis Cocker and Huey Morgan attracted 1.45 million people every week, up from 1.3m last year and 1.44m last quarter.
Overall, the BBC's radio networks reached 34.59 million people every week, up from 34.94m last quarter and 35.07m last year.
Listeners consumed more than a billion hours of radio over the quarter, with the BBC stations contributing nearly 600m of the total.
"Radio listening remains very strong across the industry with robust audiences for BBC networks," said Tim Davie, the director of BBC Audio & Music.
"After winning a Sony award for best station, these figures cap an excellent week for Radio 6 Music and represent another milestone in the growth of digital-only radio services."
Radio 4's Today programme saw its audience decline to 6.66m, down from 7.15m in the previous quarter and 7.03m in the same period last year. The station pulled in 10.31 million listeners a week in the three months, but that was down from 10.83m in the final three months of 2011 and last year.
However, digital spinoff station Radio 4 Extra, previously Radio 7, attracted a weekly audience of 1.5m listeners, up from last year's 1.16 million.
Radio 3 had a weekly reach of 1.9m listeners, compared to 2.1m last quarter and 2.26m last year, while Radio 5 Live posted a combined reach with sister station 5 live sports extra of 6.55 million listeners, up from 6.34m last quarter and 6.75m last year.
The BBC Asian Network posted a weekly reach of 540,000, up from 472,000 last quarter and 500,000 last year. Radio 1Xtra attracted 916,000 weekly listeners across the period, from 892,000 last year and 1.02m last quarter.