England declared overnight to leave Sri Lanka needing to bat all day to avoid defeat, but the home side were unable to claim the final wicket to seal the win.
The captain told Sky Sports News: "When you get so close and then get a decision which 'wins' you a Test match overturned, it's quite hard to take.
"Then for that last nick not to carry too, but it was a great Test match, and that's what cricket is all about to go all the way down to the wire, with both sides leaving everything out there.
"Those last few overs, I just thought we were going to get over the line, so it's disappointing not to. I can't praise the lads enough today. That was a flat wicket and we had to wait for the ball to reverse swing."
The second Test gets underway at Headingley on Friday.