Scott - the world number one - is two shots off the lead heading into the second round after opening his campaign at Hoylake with a four-under-par effort on Thursday.
The 2013 Masters champion started positively, posting two birdies and an eagle as he went bogey-free on the front nine.
Scott endured a tougher time on the back nine, with bogeys late in the day at the 12th and 14th holes the only blemishes in an otherwise solid round.
The 34-year-old, who surrendered a six-stroke lead as he finished second to Ernie Els in dramatic fashion at The Open in 2012, is tied for third alongside six other golfers and two strokes adrift of early pacesetter McIlroy - a position he is happy to be in.
"It's a good start, I mean I've got a long way to go," Scott told reporters.
"I have 36 holes to put myself in a position, so I'll just be grinding the next two days.
"I thought for sure this afternoon the wind kicked up a bit and the back nine played very tough.
"Coming back into it most of the time and then in and across and all those kinds of things make it very demanding but positioned it off the tee really nice today.
"I can't say it could have been better but it could have been I guess. 18 was messy but there were a couple short putts and a couple par fives.
"I didn't take advantage from a good position. But to keep pace with the early lead of Rory [McIlroy] is important I think."
Scott added: "He [McIlroy] has the potential to really put his foot down, we've seen him shoot lots of low rounds and there's nothing stopping him from doing that tomorrow either.
"I don't want him running away, we've seen what happens - he wins majors by eight.
"Martin [Kaymer] did it just a few weeks ago and it makes for a tough weekend when you're that far behind having to force it."