Jose Mourinho arrived at his former club looking to make it three victories from as many Group A matches, with a chance to bring back the feelgood factor lacking from Saturday's scoreless draw at Liverpool.
Although unable to produce a swashbuckling display, United triumphed in Lisbon as star man Rashford caught out fellow teenager Svilar with a free kick that secured a 1-0 win.
The deciding moment was a nightmare for the 18-year-old on the night he replaced Real Madrid great Iker Casillas as the youngest goalkeeper to start a Champions League match.
Only tested by a low Nemanja Matic drive in a tepid first half, United upped the ante after the break led by the former Benfica midfielder and 19-year-old forward Rashford.
England international Rashford twice called Svilar into action from corners before trying his luck with a 64th-minute free kick. The out-of-position goalkeeper backpedalled in an attempt to stop the whipped effort, but could only carry it over the line.
Rashford's ingenuity was key and Mourinho will be hoping the injury that forced him off late on is not too serious given the absentees the United boss has already.
Victory means United are on the brink of reaching the knockout phase for the first time since 2013-14, whereas Benfica have yet to muster a point at the halfway point of Group A. To make matters worse, they will be without captain Luisao at Old Trafford after his late sending-off.
Benfica's already precarious position meant it was bold to plump for Svilar in goal - a player whose unorthodox diving header clearance was the most exciting moment of a drab start.
It took 15 minutes for the first flicker of quality to bring the first chance. Benfica left-back Alex Grimaldo showed great skill and awareness when bombing down the flank before sending in a cross that Eduardo Salvio drilled wide.
Romelu Lukaku responded with a header that smashed back off the crossbar, but referee Felix Zwayer whistled for an infringement as the match returned to its plodding pace.
There was a brief period of United attacking excitement before the break as Rashford and Henrikh Mkhitaryan missed the target either side of a Matic low attempt that Svilar dealt with - the only shot on target of a dire opening period.
Ander Herrera drilled wide and Rashford had a hopeful effort when play resumed, with the latter soon showing exceptional skill to take Douglas off his feet out wide.
Matic flashed over from long range and Rashford twice threatened Benfica's teenage goalkeeper directly from corners as United pushed for the opener.
However, Rashford had found a chink in Svilar's armour and made it count from a free-kick out wide, catching the teenager out of position as the goalkeeper failed to stop the ball crossing the line.
Rashford would soon limp off to be replaced by Anthony Martial, with the in-form substitute swiftly showing great skill before having a near-post attempt stopped as the match petered out.
Luisao's second yellow card late on for a late lunge on substitute Scott McTominay compounded a miserable night for Benfica.