United summer recruits Aaron Wan-Bissaka and Harry Maguire were handed starts at Old Trafford by boss Ole Gunnar Solskjaer.
New Chelsea manager Frank Lampard named Tammy Abraham and Mason Mount in a youthful starting line-up as more established stars N'Golo Kante, Marcos Alonso and Olivier Giroud had to settle for places on the bench.
The Blues made a strong start and Abraham hit the woodwork in the opening exchanges but the hosts took the lead in the 18th minute when Kurt Zouma brought down Rashford inside the penalty area.
The England international stepped up to take the spot kick himself, after a VAR check, and confidently dispatched the ball into the top corner.
United had a Rashford goal ruled out for offside 10 minutes before the interval but Chelsea were unlucky not to go into half-time level.
Ross Barkley saw an effort well saved by David de Gea and Emerson struck the woodwork as Chelsea were denied by the post for a second time in the opening 45 minutes.
United doubled their advantage on 65 minutes when Anthony Martial bundled home an Andreas Pereira cross and the contest was effectively ended two minutes later as Rashford completed his brace.
Paul Pogba's pass sliced open the visiting defence and Rashford kept his cool to slot the ball home.
There was still time for a debut goal for James, a summer signing from Swansea, as he combined with Pogba in a rapid attack.
The Frenchman picked out the winger and James initially looked unsure as to whether to cross or shoot, with his deflected effort finding the bottom corner.
The result may have flattered United a little but there can be no denying they thoroughly deserved the three points, while Lampard may have to go back to the drawing board after a heavy defeat in the opening round of Premier League fixtures.