By Rob Dugdale
BBC Sport in Birmingham
Mensah-Bonsu was instrumental in a tight match
GB's men's team came close to losing at home to Hungary but escaped to a 66-64 win after a tense final two minutes.
Nate Reinking hit 17 points and Pops Mensah-Bonsu added 15 points and an astonishing 17 rebounds.
The win means that Chris Finch's team can win their group and qualify for next year's European finals if they win in Macedonia next Monday.
GB's women also won, 72-63 against Germany in Wolfenbuettel, with Julie Page scoring 20 points.
For once, Chicago Bulls star Luol Deng had an indifferent night offensively, missing nine shots from the field and six free throws in scoring 13 points, his lowest haul ever in a game for the national side.
GB did, however, make their most assured start to a game this summer, Reinking scoring 11 points as they took an early 15-4 lead on the way to a comfortable 37-22 half-time lead.
But Hungary started the second half with the first seven points, all from forward Peter Lorant, and a three from their guard Akos Horvath with 3:45 left of the third quarter made the score 46-41 to GB.
Free throws from Deng and Mensah-Bonsu restored a double-digit lead but Hungary were not finished.
They hit a 13-0 run in the middle of the fourth quarter as GB's offence ground to a halt for almost seven minutes and were only 63-61 down with two minutes remaining.
We'll give all the credit to the fans, because we sure didn't go out and win this one
After a Deng free throw, Adam Hanga hit a three to tie the game 64-64 with 1:26 remaining. Mensah-Bonsu hit a single free throw with 54 seconds left.
Deng then produced a defensive play to win the game, blocking Hanga's lay-up after he had seemed set to drive to the basket to give the visitors the lead with 26 seconds remaining.
Deng's free throw at the other end with seven seconds on the clock proved to be enough when Lorant missed a shot on the buzzer to tie the game.
"That was what I would call a lucky win," said Finch after the game. "If that game goes two more minutes or even two more possessions, we don't win the game. The key tonight was Pops' rebounding. His rebounding won us the game."
"We'll give all the credit to the fans, because we sure didn't go out and win this one," added Reinking. "We were just too flat in the second half so in the last couple of minutes they brought us through."
Finch told the after-match press conference that he was dissatisfied with the performance, but admitted some of the blame, claiming he had been a little soft on his players. That will change, he said, in the four days before the team flies to Skopje for the vital game with Macedonia.
"We'll be working on toughness," he said. "We know our plays now, we just have to execute them through the resistance of someone that doesn't want you to do it."
"If it's Macedonia or whoever, we have to get our edge back and come at them," added a subdued Reinking.
GB's women rebounded from the home defeat against Slovakia to lead almost from tip to buzzer against Germany. Page had an inspired night, hitting 12 of her 20 points in the second half, while Natalie Stafford weighed in with 16 points.
The team survived Kim Butler fouling out in the early third quarter to establish a 10-point cushion that their hosts only threatened in the last five minutes before Stafford's three with 1:18 remaining killed the game.
"We kept them to 63 and doing that you expect to win," said coach Tom Maher. "We had a really good start and even down the stretch when we couldn't make any foul shots, we didn't toss the ball away."