"We were delighted to get seven goals but we had to work very hard for it," said the Rangers manager.
"Falkirk play some good football, they don't sit back and defend.
"At half-time it was a tight game. We managed to play better in the second half, our possession of the ball was better and we had far more opportunities.
"But you have to look at Falkirk and the way they play - you have to admire the way they play. They don't come to Ibrox to sit back and defend, they come to try to play football."
Rangers headed into the game boasting five consecutive clean sheets but Arnau Riera and Graham Barrett were both on target for Falkirk and Smith added: "We lost two goals today - the first we have lost this season.
"There was always the fear they could get back into the game. We had to play extremely well to add to the scoreline that we had and we had to work very hard to do it."
The flurry of goals allowed Boyd to get off the mark after a troubled start to the campaign.
The striker has been roundly criticised for his lack of prowess in front of goal and was named on the bench again today after his role as an unused substitute against Red Star Belgrade in midweek.
He replaced Lee McCulloch, who complained of feeling unwell, and answered his critics by contributing to the rout with a goal as well as an assist.
"Some strikers go through those difficult spells," Smith shrugged. "But he has shown over the last few seasons that he will score goals.
"He scored today and he will score in other games for Rangers this season as well."
Smith also had another pop at the under-21 rule, which forces managers in Scotland to name three youngsters on the bench.
"The problem we have in Scotland is that we have to leave players out," he said.
"Brahim Hemdani and DaMarcus Beasley play in a team that wins 1-0 in the Champions League and then they can't even get a place on the bench on a Saturday.
"It's a ludicrous situation and one that makes it very difficult for everyone to try to get a group of players together.
"But, while it continues, we have to handle it."