Darren Bent's opener was quickly cancelled out by Robert Snodgrass in the closing stages of the first half before Stockdale made a miraculous one-handed save to deny Ricky van Wolfswinkel from six yards out.
Norwich had the better of the chances, but now face a replay at Craven Cottage and Meulensteen was pleased with Stockdale's intervention.
The Dutchman said: "We didn't want it (a replay), I think Chris (Hughton, the Norwich boss) didn't want it either, but that's how it turned out. A draw's a good result away from home.
"David Stockdale had a Gordon Banks-like save to keep us in the game.
"It was definitely one of the best saves I have seen. It was a fantastic cross by (Javier) Garrido and a great leap by Van Wolfswinkel.
"I honestly thought it was in but it was a fantastic save. That's what makes football so exciting sometimes."
Hughton praised Stockdale for the stop.
"Outstanding," the Norwich boss said. "From the angle we had we thought it was in. It's a wonderful save."
Hughton lamented the slack defending which allowed Bent to head Fulham in front, when goalkeeper Mark Bunn was caught in no-man's land trying to meet Chris David's cross.
"We conceded a poor goal," Hughton said. "It's there for all to see - it was a fairly hopeful ball into the box and we should have dealt with it better.
"We showed good spirit to come back and scored the goal at the right time.
"We gave everything to try to win the game. On the balance of play, I thought we deserved it.
"It's a replay we would rather not have, but having said that we would rather still be in the competition than out."
Injuries to nine first-team players necessitated changes for Norwich and allowed 18-year-old Josh Murphy the opportunity to impress on his debut on the left wing.
Wes Hoolahan was among those absent amid reports of a move away from Carrow Road, speculation Hughton dismissed.
The Norwich boss added: "Wes reported an injury yesterday and at this moment is having treatment. There has been no bid that I am aware of, absolutely not.
"It's not our intention to lose players, our intentions are to strengthen what we have got here and to make sure we get these players back fit."
Fulham talisman Dimitar Berbatov was also missing, but Meulensteen insisted the sole reason for the striker's omission was to allow him to recuperate following a recent injury.
The Fulham boss added: "He's not injured but because of him coming back from injury and being able to play the 90 minutes against West Ham, we thought it was better for him to recuperate and be ready for the Sunderland game."