But QPR sealed a 2-1 win after restoring their lead in the 63rd minute when the ball ended up in the back of the net as Clint Hill, Jenas and visiting defender Jake Buxton challenged for Alejandro Faurlin's right-wing free-kick at the far post.
That condemed Derby to their first defeat since McClaren's appointment, while it was also promotion-chasing Rangers' first win in four matches and first victory against a top-nine team this term.
However, their afternoon took a worrying turn for the worse when fans favourite Faurlin, who suffered a long-term knee injury the season before last, was carried off on a stretcher off with 15 minutes left.
QPR went ahead after some good work on the right by summer signing Matt Phillips, who later hit the woodwork.
The winger has struggled for fitness and form since his move to west London from Blackpool, but produced a fine display and set up Rangers' 11th-minute opener with an excellent cross.
Phillips found Charlie Austin and after the striker's close-range effort was kept out by keeper Lee Grant, Jenas had the simple task of scoring into an unguarded net.
But the visitors hit back in the 23rd minutes when Dawkins volleyed in after Jamie Ward's free-kick had been headed down by Buxton.
It was Dawkins' first goal for Derby since McClaren signed him on loan from Tottenham and it angered the hosts, who felt Buxton committed a foul as he climbed between Austin and R's defender Richard Dunne.
Austin headed wide from Faurlin's corner and Junior Hoilett fired over as Rangers looked to regain the lead, but Derby remained a threat and almost scored again early in the second half.
Dawkins robbed Danny Simpson near the edge of the penalty area and teed up Ward whose shot was parried by keeper Rob Green, with Chris Martin unable to force home the rebound from an awkward angle.
Having caused problems on the break and defended well for long periods, Derby found themselves behind again following Faurlin's wicked delivery which led to the winning goal.
And Phillips almost added a brilliant third, cutting in from the right and evading two challenges before sending a rasping shot against the underside of the bar.
Green produced a fingertip save to keep out Ward's injury-time free-kick and ensure the R's held on for a deserved three points.
But they will be anxiously waiting for news on Faurlin, who was unable to continue after a challenge on John Eustace near the halfway line.