Murray, making his second debut for the Cumbrians, after an 11-year absence, pounced after just four minutes to volley into an empty net from Karl Hawley's pinpoint cross.
It was the dream start for last season's League Two champions but after making the early running they only clung on to take the points amid an hour-long barrage of Rovers pressure.
Paul Heffernan should have equalised for the visitors a matter of seconds after Murray's goal but despite expertly creating space for himself he blasted straight at the onrushing Keiren Westood.
Captain Kevin Gray almost doubled Carlisle's lead with a backpost header before Hawley and Kevin Gall both drilled shots inches wide.
Rovers chief threat Jonathan Forte was tormenting debutant full-back David Raven and it was no surprise when the ex-Liverpool youngster limped off after just 28 minutes.
As Carlisle eased into their comfort zone, Rovers pushed on with James Coppinger and Graeme Lee both failing to find the target.
The majority of Carlisle's huge opening day crowd, 9,433, breathed a sigh of relief early in the second half when Bruce Dyer ran clean through only to head tamely at the grateful Westwood.
Lee smashed a clearance up to Dyer and his marker Gray, but when the veteran defender fell flat on his face Dyer was left with just the keeper to beat but succeeded only in helping the ball into Westwood's hands.
Rovers were by now completely dominant and only a first-class save from Westwood kept the frustrated Dyer at bay.
When Westwood was finally beaten, from another Lee header substitute Paul Arnison came to the rescue with a diving header off the line.
Doncaster failed to make the most of their late possession as they ran out of steam leaving Carlisle thinking that a third straight promotion could be on the cards.