The League One strugglers advanced to the third round after coming from behind to snatch a famous 3-2 victory at a jubilant Stair Park.
Martin Grehan grabbed the winner eight minutes from time, shortly after Orhan Mustafi's close-range finish looked to have forced extra-time.
Adams said: "We go in 1-0 at half time and are in control of the match at that stage. To get back to 2-2 with I think five minutes to go, you think you can get them to extra time.
"Obviously, they scored the goal and were able to go through. I've been in cup shocks myself and it is always good for them but for us we are obviously disappointed.
In contrast, Stranraer counterpart Stevie Aitken rated the achievement as the biggest of his 10-month managerial career.
He said: "It's up there - it must be - to beat a Premiership team and a good established Premiership team, it must be up there.
"I'm over the moon, I'm delighted, and I thought at half-time that my talk to you (the media) after the game would be something different.
"But we will concentrate on the players - I thought they were immense.
"My worry at half-time was we would come in deflated because of the penalty decision against us.
"I think in the first half we showed up really well and I worried a wee bit about that. I knew and I could sense the players were in the game.
"The next goal was always crucial. If they got it then it would be difficult for us but we got it.
"I thought we kicked on from there. I thought the penalty we got was soft as well to be honest - great finish though. I thought the third goal was great; a great move."