Chasing 355 to stay alive in the series, Pakistan were on the hunt with Saud guiding the hosts to 291-6 in the morning session on day four.
Saud was 94 when he gloved a Mark Wood delivery down the side of the leg and Ollie Pope dove to his right to collect the ball.
On-court umpire Aleem Dar ruled the batsman caught late in his soft signal before returning the decision to television umpire Joel Wilson.
Replays could not confirm whether it was a clean catch, and the original decision was upheld at a critical time in the seesaw contest.
“Shakeel’s dismissal cost us,” Babar told reporters after England won the series with one game to spare.
“It looked like the ball had hit the ground.
“As a professional, you have to respect the referee’s decision, but we felt the ball went down.”
Pope also had no ambiguity about the hold he took to turn the game around.
“I didn’t think it had bounced back before,” the 24-year-old said.
“As a goalkeeper, honestly when you’re wearing gloves you can’t tell. I didn’t think he hit the ground afterwards. When you’re wearing gloves you feel him coming in, but that’s it. .”
England recorded, with one game to spare, their first series win in Pakistan since 2000-01 and captained Ben Stokes put the achievement into perspective.
“Coming to the subcontinent, it’s always a tough place to come and win,” Stokes told Sky Sports.
“We know what we’ve achieved this week. We know it’s something that’s not unheard of, but very rare to do, especially as an England team.
“We understand how special this achievement is this week, but as we say over and over, these streak wins and wins are part of a much bigger picture of what we’re trying to achieve right now.”