Anything other than an England win would have seen Jos Buttler's men knocked out and they were left fearing an early exit when rain delayed the scheduled start by three hours.
England, in a match eventually reduced from 11 to 10 overs per side by a fresh downpour, collapsed to 13-2 after losing the toss.
But a stand of 56 between the Yorkshire duo of Harry Brook (47 not out) and Jonny Bairstow (31) revived England's innings.
Namibia were left with a target of 126 under the Duckworth-Lewis-Stern method for rain-interrupted matches. But the non-Test nation were never up with the required run-rate and finished on 84-3.
Victory took England level on points with Scotland in Group B but with a superior net run-rate -- the tie-breaker for teams on equal points.
England, however, could still be knocked out should Scotland achieve a stunning upset win over already-qualified Australia in the final Group B fixture in St Lucia later Saturday, or if that game ends in a no-result.
Earlier, Ruben Trumpelmann bowled England captain Buttler, one of the world's best white-ball batsman, for a four-ball duck as he nipped one back sharply off the pitch.
Veteran all-rounder David Wiese then produced a magnificent slower ball to have Phil Salt, England's other opener, caught behind.
Rain stopped play with England 82-3 off eight overs.
When the match resumed, Brook did the damage in an over from Namibia teenage paceman Jack Brassell, which cost 19 runs, before Moeen Ali and Liam Livingstone struck three sixes between them in the last over from Trumpelmann (2-31).
The already-eliminated Namibia retired opener Nikolaas Davin out for 18 to allow the big-hitting Wiese to come in at 44-1 off six overs.
But even with the 39-year-old Wiese -- who holed out in the last over of his final international appearance -- hitting two sixes and two fours in a 12-ball 27, Namibia finished well-short of their target.
