The concession of a 72nd minute Christie Elliott equaliser following a dominant first period - during which they only had a headed strike from captain Christophe Berra to show for their command - means they will have to face the Maryhill side in a replay at Tynecastle next Tuesday.
Levein would not be drawn on the semi-final draw, which pairs the winners of the replay with Inverness Caledonian Thistle at Hampden, as offering any added incentive in now ensuring only two Ladbrokes Championship teams stand between his club and a first final in the competition since their 2012 triumph.
The Tynecastle manager’s focus was on his belief that Madden judged Uche Ikpeazu on his size and physicality when he went to ground under a challenge from Gary Harkins after 77 minutes.
“I really feel sorry for Uche,” said Levein. “Just because he’s stronger people shouldn’t be allowed to foul him. I don’t care what way you go down if you get fouled it’s a penalty.
“Take that away, we still had other chances to kill it. I don’t want to start telling him to go down. He’s a strong boy and he should get an even break.
“If he’s stronger it’s not a foul but he constantly gets pulled up because he’s stronger than others. I think there’s nobody else in Scotland like him so he’s different and he gets penalised for being stronger.
“[Yet] we did enough to win in the first half and didn’t get that gap with the second goal which would have been game over.
“Credit to Partick, they kept plugging away and caught us on a counter which was annoying because we’re usually good at dealing with that. Their goal was from probably their first shot on target.
“They’re disappointed but our intention was to come out of the game still in the draw and we still are in the draw.
“We’ve got a game back at Tynecastle which I feel confident about so it’s not a bad situation for us.”
His Thistle counterpart Caldwell praised his team’s “courage and bravery on the ball” to extricate themselves from a tie that looked to be too much for them in the early stages.
“It was big disappointment first half,” he said. “We asked for composure and passing but we didn’t do it until 25 or 30 minutes in. We knew we’d have to defend and deal with Hearts’ physicality at set pieces. “They have great variation and we did that very well apart from the goal. Once we grew into the game in the second half I thought we were outstanding.
“We played well, passed the ball well and scored a brilliant goal. We played in their half more and had territory higher up the pitch.”