The centre-back, who signed for the Blues in January from Bolton Wanderers for an estimated £7 million, was handed his first start at Stamford Bridge but the match finished 3-3 after the hosts threw away a 3-0 advantage.
The 26-year-old admitted that the result was far from what he would have liked on his debut, but explained that with defenders in makeshift positions it could have gone worse.
He told the Chelsea’s official website: “It was an amazing feeling to pull on the shirt and play my first game for my new club.
“It was a fantastic occasion and there aren't many bigger games I could have come into for my first one. As I've said before, the reason I moved here is to be involved in games like that so it was brilliant.
“I think playing with David Luiz worked really well, he's a top-class centre back. Obviously with Jose Bosingwa playing at left-back, which he probably hasn't done for absolutely ages, and David Luiz moving from the right to the left in the middle it was all change, but it worked okay.
“They scored two penalties which we couldn't really do anything about, but I said when I arrived that there are top-class defenders at the club, and it's easier to come into a new side and play alongside players of such a high standard.”
Goals from Luiz, Juan Mata and an own goal from United defender Jonny Evans set the home side up for a dream result but two Wayne Rooney penalties and a late header from Javier Hernandez earned the visitors a draw.
Cahill revealed: “I think it's been put out of our minds now, but it was a disappointing result, make no mistake about that.
“I sat there for about two hours after the game and I was wondering how it had happened, but nevertheless we didn't lose the game and there's no point in us harping on about it, we're looking ahead to a tough game at Everton on Saturday now.
“It's massively important to get back to winning ways, I think all the players are fully aware of that, we need to get back to winning games and I don't think we're too far away from that.
“Hopefully we can start against Everton, but all the lads know what we have to do.”
The west London side travel to Goodison Park on Saturday and will be keen to avoid the same fate league leaders Manchester City suffered on Merseyside.
Cahill added: “I noticed myself that all of the teams around us have got tough games this weekend, but we can't do anything about that, that's out of our hands.
“We can only influence the game we have at Everton; we've had two good results against them already this season so hopefully we can make it a third.
“Having drawn our last three games we need a win, it will be tough as we saw what they are capable of when they beat Manchester City last week so we will need to approach the game in the right way.”
The England international stressed the threat Everton pose, insisting they have become even more of a danger over the past few weeks.
“It's always a tough match at Everton, I played them just before I moved here, it was my last game for Bolton,” the former Aston Villa defender added.
“In the time since then their team's changed a bit, and with the likes of Pienaar and Gibson in the side they are definitely stronger, it will be tough for sure.
“I hope it goes exactly how it did last time I played there; we won the game and I scored, you never can tell obviously, but that would be the ideal scenario.”