A first half penalty from Joe Nuttall was enough for Rovers to make it four consecutive wins in all competitions and start 2019 in perfect fashion.
Nuttall, who is in fine form for the Development Squad, calmly slotted home from the spot just three minutes before the break after Brad Lyons was tripped in the box by Marc Guehi.
Played at Aldershot Town's EBB Stadium, Rovers made a good start, with Nuttall proving a nuisance for the Chelsea backline in Hampshire.
Nuttall and John Buckley tested Jamie Cumming early on, before Stefan Mols cracked a shot against the upright later on.
At the other end, home attacker Tino Anjorin worked a chance, which Tyler Magloire managed to block well to keep the scores at 0-0.
Tariq Lamptey skipped away from the returning Matty Platt but saw Ollie Byrne in his way as the hosts came again, looking for the opener.
Rovers went close again when a Nuttall header was flicked onto the underside of the crossbar and away, just before the penalty was awarded, which was confidently tucked away by the in form striker.
That gave Rovers the lead at half time before Chelsea came flying out of the traps after the interval, with Anjorin forcing a couple of excellent blocks from visiting defenders.
Rovers were having to withstand heavy pressure from the hosts, but were still a danger on the attack, with Nuttall once again forcing Cumming into another save midway through the half.
Damien Johnson's side were defending excellently, and Byrne was on hand again late into the game to stop Anjorin from levelling late on, preserving his clean sheet and the points for Rovers, who move closer to safety in Premier League 2 with the narrow win.
Chelsea: Cumming, Lamptey (Nartey, 81), Colley, Guehi, Grant (c), Castillo, McEachran, Gallagher, Gilmour (Brown, 72), Anjorin, Redan (Taylor-Crossdale, 72).
Unused substitutes: Askew, Masampu.
Booked: Gallagher, Lamptey.
Rovers: Byrne, Platt, Thompson, C. Doyle, Magloire (c), Grayson, Carter, Buckley, Nuttall, Mols, Lyons
Unused substitutes: Winterbottom, Simmonds, Rankin-Costello, Paton, Vale.
Goal: Nuttall pen, 43.
Booked: Magloire, Lyons, Nuttall, Carter.
Referee: Gary Parsons.
Image: Chelsea FC