The Rams recorded a second successive away victory in the Championship and left the Glovers with plenty of food for thought.
The visitors started brightly and came close to snatching an early lead in the second minute when Martin fired narrowly over debutant Wayne Hennessey's crossbar.
Derby threatened again moments later and should have scored a deserved opener when the lively Russell found space at the edge of the box, but his shot crept wide.
Yeovil manager Gary Johnson left regular strike duo Paddy Madden and James Hayter on the bench and deployed a midfield five in an attempt to contain the Rams.
However, his side struggled to seize the initiative for large periods and often found lone striker Michael Ngoo isolated in the final third.
As the half progressed Rams' midfield trio Jeff Hendrick, Craig Bryson and Will Hughes began to dictate proceedings with their movement and sharp interchanges causing increasing concern amongst Yeovil's rearguard.
That concern was compounded four minutes before the break when Richard Keogh found summer signing Russell, who peeled off his marker and expertly curled the ball past Hennessey from close range.
The visitors doubled their lead two minutes later after a scintillating move by Russell and Martin put through midfield talisman Bryson, who effortlessly slid the ball into the net.
A shellshocked Yeovil were left hanging on for half-time and should have found themselves 3-0 down at the break after Jamie Ward, who was restored by manager Nigel Clough into the Rams starting line-up, saw his right footed strike smartly saved by Hennessey.
The hosts brought on last season's League One golden boot winner Madden and new signing Andy Williams at the break but it was Derby again who looked the most threatening.
Despite an improved spell at the start of the second period, Yeovil's dismal day continued on 58 minutes when Ward's 25-yard strike was palmed away by Hennessey, only for the onrushing Martin to score his side's third goal.
The Somerset side, whose only league goal came courtesy of Ed Upson's late winner at Millwall on the opening day of the season, poured players forward at every attempt and came close to pulling a goal back when winger Liam Davis' ferocious effort was smartly pushed away by Lee Grant, before Williams saw a penalty appeal turned down by referee Keith Hill.
A flurry of corners followed as the hosts looked for a consolation, but the Rams defence held firm to cap off an impressive away performance.