Hearts manager Daniel Stendel admitted Falkirk made life difficult but felt his team deserved to reach the Scottish Cup quarter-finals.
Sean Clare's 50th-minute penalty-kick settled the fifth-round tie after Liam Boyce was fouled by the Falkirk full-back Michael Doyle.
The hosts then struck the post twice through Declan McManus before Gary Miller hit the crossbar. However, Hearts regained composure to see the game out despite the inclement effects of Storm Ciara.
"It was not so easy today but in the end it was important that we won this game," said Stendel afterwards. "In some situations we had a bit of luck but we missed that sometimes in the last games.
"Falkirk had some chances but you could see we were the better team. We scored, it was a penalty, it doesn't matter. We scored and we won the game. This is important for us.
"The weather was crazier before the game but I told the players it did not matter. Falkirk started as we expected, very hard. We lost our confidence a little bit in the first half, we lost the ball too easily and gave Falkirk a chance to come into the game.
"That was what we did not want to do but in the end I am happy about the result. We are looking forward to Wednesday."
Hearts travel to Celtic in the league on Wednesday and must wait to learn if centre-back John Souttar will be fit. He sat out the Falkirk tie due to an ankle injury.
"After Wednesday's game he was not fit to play. I hope he comes back next week. We have big games and we need him," stated Stendel.
Falkirk's co-manager David McCracken could only lament the chances his team created without converting. After falling 1-0 behind, McManus struck two shots inside two minutes which both careered off the right-hand post of Hearts goalkeeper Joel Pereira. Then Gary Miller dispatched a 25-yard effort off the bar.
"We're gutted because over the piece we created a number of opportunities," said McCracken. " The penalty goes against us and it's disappointing but a big positive was how we reacted.
"We created opportunities and it's a fine line when it comes to taking them. You start to think: 'Is one going to go in?' Unfortunately it didn't. We didn't sit in and defend. We tried to win the game but ultimately we're out of the cup."