Billy McKay nodded home a late equaliser to earn his side a replay minutes after Jamie Longworth had given the Stair Park side a shock lead.
Longworth kept his cool to net his 20th goal of the season after Martin Grehan had equalised from the penalty spot on the stroke of half time.
Aaron Doran had put the League Cup finalists ahead with a curling effort in the first half but a replay was no less than Stevie Aitken's men deserved.
Caley Thistle boss John Hughes warned his team they could still fall victim to a cup shock.
He said: "The tie is not over. They are a right good team. They try and pass it, they work their socks off and we will need to make sure we are at our best to make sure we're in the hat for the next round.
"There was a bit of mental fatigue out there. We will have to give them two or three days off now to recover. Games are coming thick and fast.
"It would have been a massive come-down if we had lost here. We pulled it out of the hat after giving them two goals but we showed character to get back.
"I felt the boys gave me everything they had. I told them there would be a game on here but we've still got work to do."
Stranraer manager Stevie Aitken was delighted with his team's workrate and was desperate to win the game on the day to avoid a midweek trip to the Highlands.
He said: "Midweek games are very difficult. Some of the boys will need to take two days off work for it.
"I tried to go for it at the end. I tried to fling as many men forward just to get that win but if somebody had said to me before the game it would be a draw I would have took it.
"We've got a confident bunch of lads. Every time they go out on the park they believe they can go and do something. I think that showed today.
"Against Premiership teams you sometimes only get one crack at it. We will go up there and give it our best shot but they are a very good team."