England appeared to be cruising during a 98-run stand between Morgan (61) and Luke Wright, despite already having stumbled to 103 for five at one stage in pursuit of an under-par 241 at the Brit Oval.
But Umar Gul snared Morgan, caught in the leg-side deep, with the first ball of his third spell - on his way to match-winning, career-best figures of six for 42.
The new batsmen then failed to cope with Gul's reverse swing, allowing Pakistan to bowl England out for 218 and pull off an unlikely, but in the end comfortable, win which makes the one-day international series score 2-1 with two to play.
Strauss, who himself made 57 at the top of the order, gave Pakistan credit but felt his own team had let an opportunity slip to wrap up the series early.
"It leaves a bit of a sour taste," he said.
"We are very disappointed with the way we chased those runs. But Pakistan are capable of putting you under pressure with the ball - and they were able to do that.
"Eoin Morgan and Luke Wright looked comfortable - it just goes to show how the game of one-day cricket can change very quickly.
"Chasing 240, you back yourself to win at The Oval. But it was an outstanding spell from Gul.
"He got the ball reversing both ways, and that is always tough to deal with.
"You don't want to be in a position where a bowler gets six wickets.
"We will do a debrief and try to learn the lessons, but you have to give credit to the way Pakistan defended that total."
Strauss was pleased, however, with England's display in the field - in particular James Anderson's three for 26 against opponents for whom only Fawad Alam (64) could pass 50.
"The fielding was much improved, and Jimmy was outstanding.
"To restrict a team to 240 at The Oval, you think you are in the game.
"We are disappointed but we are still leading the series 2-1 - and we hope we can win the next game at Lord's.
"We started losing wickets quickly and often - if we had batted 50 overs we would have won."
Pakistan coach Waqar Younis unsurprisingly had a similar take on the ebbs and flows of an intriguing contest.
"There was no way for us to bowl 50 overs and win this game," he said.
"Our only chance was to bowl the English team out - and we've done that."
Gul was almost single-handedly responsible.
"After the two losses, we really needed something like this - and I'm glad the way it came, especially with the way Umar Gul bowled," Waqar continued.
"It was outstanding
"We could maybe have got a few more runs.
"But Shoaib Akhtar was important too, because we really needed early wickets."
Waqar hopes Pakistan can dig deep again for a deserved reward for somehow keeping their focus on cricket at the end of a long and arduous tour - which will inevitably be remembered for all the wrong reasons thanks to the spot-fixing controversy of the past three weeks.
"This wasn't a fluke, because the boys really played well in a team effort," he insisted.
"The series is still alive, and this gives us the incentive that we can come back and maybe win the series.
"I suppose it's maybe a bit early to say that, but there's a chance now.
"It's been a really tough tour on the field and off it.
"We're pretty pleased to be back in this one-day series, but there's still plenty of work to be done."
Pakistan wicketkeeper Kamran Akmal was expected to have an x-ray tonight on the finger injury which forced him off the field.
