McLeish was greeted with boos and cries of 'you don't know what you're doing' when he took off Charles N'Zogbia and replaced him with Weimann in the 73rd minute.
But with seconds remaining, Weimann earned Villa their first home win for four months after fellow substitute Gary Gardner's shot had been parried by Mark Schwarzer.
McLeish said: "Listen, the fans were great. They see a forward coming off and Charles has been in good form, no doubt about it, but I did want to get Weimann onto the pitch at some stage.
"I'll take it (the boos) on the chin. You don't want to be hearing it but it is part of the modern game.
"The fans were excellent but I thought we gave the fans something to be excellent about because of the way we played."
Austrian Under-21 international Weimann has been pushing for a first-team spot after some prolific form for the reserves.
McLeish said: "Sometimes the subs work for you. Weimann was a certainty to come on.
"I discussed that with the coaches beforehand and said somehow we've got to work this boy into the team if it's 0-0 or we are a goal down because he is a goalscorer.
"It is just trying to fit him into the team at the right time. He was competing against guys like Darren Bent and Gabriel Agbonlahor.
"Now there are a couple of injuries to key players and Andreas had deserved his moment.
"He has been prolific in the reserves, he is hungry, he wants to score goals, he links up play as well. He is not just a penalty-box player."
McLeish was delighted with his side's general performance against a Fulham side which had won their previous three games.
He said: "Today [Saturday] we played some great football and it is being clinical in that final third. That is the difference between us climbing up the table a few rungs.
"What makes it so satisfying is the win over a team of high quality that we deserved to beat.
"It is a great win, we took a bit too long to score the winner in my opinion, but it was deserved."
Fulham boss Martin Jol admitted: "It was a real kick in the teeth and harsh to concede in the final minute.
"I don't blame anyone for the goal but I feel in the last minute you have to keep the opposition in front of you. We didn't do that for the goal.
"Villa pressed us high up the pitch in the first half and did well but in the second half we were the better team.
"We changed the system, created a spare player in midfield for the second half and that was the solution for us."
Jol took the decision to replace skipper Danny Murphy at half-time after he received a cut to his face in an aerial challenge with Stephen Ireland.
He said: "Danny had a cut to his face and they had to stitch it so I made the decision to put someone else on instead of waiting for the doctor's decision."