The Red Devils striker went down after being caught by the trailing leg of Blues debutant Cahill at Stamford Bridge, prompting an outraged Sir Alex Ferguson to race onto the pitch as referee Howard Webb ignored the United appeals.
The England international feels he won the ball before he clipped Welbeck but admits he was lucky the incident went unpunished due to the nature of the challenge.
"I thought the contact was outside the box anyway, and at the time I thought I had got the bottom of my studs on the ball," Cahill, a January arrival from Bolton, told Chelsea's official website.
"I have seen the replay and sometimes when you go through the player and across the player to get the ball they are given against you, so I was fortunate to get away with that one but if you see the direction the ball moves I think my studs actually touched it first."
The incident was one of two United penalty cries ignored in the first half after Ashley Young had gone down easily under pressure from Jose Bosingwa.
But the visitors were awarded two spot-kicks after the break as they fought back from 3-0 down to salvage a draw, with Wayne Rooney converting both before Javier Hernandez grabbed a late header.
Cahill, whose first appearance in a Blues shirt saw him partner David Luiz at centre-back in the absence of injured captain John Terry, admits the draw felt like a defeat after throwing away such a decisive lead.
He said: "It was a fantastic game to be involved in and it was in at the deep end but these are the games you want to be involved in, however any game you go 3-0 up you expect to win so that has put a dampener on my debut.
"It is a sickening feeling like we have lost the game. The mood is a bit flat and we are disappointed.
"The second half we started fantastically well and went 3-0 up and really should have seen the game off from there, but the second penalty was disappointing.
"The first one was probably a penalty but the second one was right in front of me and I didn't think it was a penalty at the time and seeing the replay I haven't changed my mind, so that went against us at a crucial part of the game."
Cahill came close to playing out a dream debut when he almost fired home a winner for Chelsea in the final minute, but his long-range shot was tipped over the crossbar by an impressive David De Gea, who also denied Juan Mata late on.
The defender added: "I think if that was anywhere but straight down his neck then it was going in but he has pulled off two fantastic saves near the end.
"The free-kick from Juan Mata he couldn't have put it anywhere closer to the top corner and he [De Gea] has managed to claw that one out as well to get them a point."