"I thought it was the least we deserved," said the Hibs boss after the 2-2 last-16 draw at Tynecastle.
"The contribution we made, if we had lost it wouldn't have been a fair reflection of the game.
"The lads believe in themselves, and when you have that and you have team spirit, anything is possible."
Hearts were two goals up at the interval thanks to strikes by Arnaud Djoum and Sam Nicholson.
But Hibs rallied and scored in the 80th minute through Jason Cummings and forced a replay at Easter Road when Paul Hanlon slid in to score.
Stubbs told BBC Scotland: "The lads have shown great character again.
"I never doubt this team. I always felt once we scored that we were going to get the second.
"You only have to look at the amount of possession and second balls that we had in the second half. We've come here and really took the game to a really good Hearts team.
"We've managed to keep the tie alive. We're in the hat tomorrow.
"We'll see who ourselves or Hearts could come up against. There were two good teams out there today. I'm sure it will be a really close game at Easter Road."
The Edinburgh pair will be joined in Monday evening's draw with Celtic, who defeated Lowland League side East Kilbride 2-0, Morton, Dundee United, holders Inverness, Ross County, plus Kilmarnock and Rangers, who drew 0-0 on Saturday, and Dumbarton and Dundee, whose tie was also goalless.
Hearts head coach Robbie Neilson preferred to savour the prospect of another match against their old foes rather than dwell on having been pegged back.
"In the second half we handed Hibs the initiative by not pressing as well as we should have, but it's a replay, it's another money-spinner for both teams and it's going to be another huge game," he said.
"It was a great game. It wasn't a pretty football match - it was competitive and I'm sure everyone enjoyed it."
Asked if he thought Hibs were now favourites to reach the quarter-finals, Neilson said: "No, I don't think so. They've got two games before that.
"We'll have (Igor) Rossi back in, hopefully (Alim) Ozturk will have recovered from his hamstring problem so it gives us the initiative. It's going to be another good game."
Hibernian's dramatic comeback against Edinburgh rivals Hearts to force a Scottish Cup replay was thoroughly merited, claims manager Alan Stubbs.