Namibia reached 34-0 off five overs, the minimum number required, but the run rate required was far too much even after Nikolaas Davin (18 off 16 balls) was retired out to let in David Wiese who made 27 off 12 balls.
Namibia ended their innings on 84-3 before captain Gerhard Erasmus confirming that the match at the Sir Vivian Richards Stadium in North Sound was Wiese's farewell.
England up to second in Group B, ahead of Scotland on net run-rate but level on five points, with their rivals set to play leaders Australia later on Saturday.
A draw or win for the Scots in St Lucia would see them through at England's expense.
"There was a lot of anxiety going around," said Brook of the long wait for the game to start.
"We didn't quite think we were going to get out there. But thankfully it (the rain) stopped. We took quite a bit of information from our batting out onto the pitch when fielding."
England made a shaky start with Phil Salt getting just one run on the board in the first over and captain Jos Buttler departing for a duck, bowled by Ruben Trumpelmann in the second.
It was 13-2 after 2.1 overs when Salt was caught by wicketkeeper Zane Green off Wiese but Jonny Bairstow, also caught by Green for 31 off 18 balls, and Brook (top scorer off 20 balls) steadied the ship with a 56-run partnership.
The match was halted again by rain after eight overs, with England on 82-3, but quickly resumed.
"It's a big relief. It's certainly a stressful day when the rain was coming down, but I thought we put in a really good performance. Everyone managed that situation well, we were well prepared when we did play," Buttler said.
"I thought that was a really good score on that wicket."
England's opening game against Scotland in Barbados was washed out and the holders lost by 36 runs to Australia before thrashing Oman by eight wickets.
That victory gave England their crucial superior net run-rate over Scotland but they needed to despatch Namibia to make it count.