Craig Bryson put the home side in front after just three minutes and Jason Pearce hit the bar for Portsmouth before Chris Maguire scored his first league goal for Derby to double their lead.
Jamie Ward made it 3-0 after 33 minutes with a precise finish and Portsmouth rarely looked like mounting a comeback.
David Norris headed wide at the start of the second half and Pearce pulled a goal back with a late header but Derby held on for the three points to move up to fifth in the table.
Derby got off to the perfect start with a goal after three minutes from Bryson, who controlled Theo Robinson's pass on the edge of the box before producing a fine finish into the bottom left-hand corner.
Portsmouth came close to drawing level with 11 minutes gone when Lawrence's corner picked out Pearce but his header cannoned off the bar.
The Rams compounded Portsmouth's misery three minutes later by doubling their lead when Stephen Henderson parried Robinson's shot and Maguire was on hand to fire home the rebound.
Derby immediately went in search of a third goal and Jeff Hendrick had a shot saved by Henderson before Bryson fired wide with an ambitious attempt from 30 yards.
The home side suffered a blow when Mark O'Brien was forced off and replaced by Kevin Kilbane for the Irishman's 600th club appearance.
The enforced changed did not affect Derby's rhythm, however, and Ward let fly from 25 yards but his shot was tipped over by Henderson.
With 33 minutes gone Ward did get the better of Henderson with a fine shot into the bottom right-hand corner from 18 yards to virtually make the game safe for Derby.
It could have got worse for Portsmouth but Henderson tipped a shot from Maguire over the bar before denying the Scot again minutes later with another fine save.
Robinson broke down the left and gave the ball to Bryson who laid if off to Maguire but Henderson was off his line quickly to block the shot.
In first-half injury-time Portsmouth could have given themselves a lifeline but Dave Kitson headed wide from another pinpoint Lawrence corner.
Derby took their foot off pedal at the start of the second half and they almost paid for it six minutes after the break.
Lawrence curled a superb ball into the box and Norris got in at the near post but his header went narrowly wide.
That let-off woke the Rams up and Robinson had a low shot well saved by Henderson.
With 11 minutes remaining, Portsmouth worked a corner well and Lawrence's ball in found Pearce who nodded in from close range but Derby saw the game out.