Rangers manager Walter Smith was delighted with his team's triumph.
"It was always going to be a difficult match for us today the way Falkirk set out their team.
"Today, we never reached the levels we did do last week.
"The goal from Nacho Novo was terrific for us.''
Smith admitted Rangers might have suffered a bit of a hangover after winning their first league title for four years last weekend.
"It was going to be a bit of a struggle to get everybody back up from that fantastic effort from us last week,'' said Smith, who refused to speculate on the future of midfielder Barry Ferguson.
"I think you could see the game did take its toll.''
Gers' assistant manager McCoist applauded Falkirk's performance.
He told BBC1 Scotland: "I think they made difficult for us throughout the game.
"They didn't have a lot of chances at our goal and they only hit the post at the end there. I think Neil McCann hit the bar in the first half.
"I think overall we're just delighted to get the win. I'm chuffed to bits for the team.''
McCoist also hailed Nacho Novo's 35-yard volley as one of the all-time great cup final goals after the Spaniard sunk Falkirk at Hampden.
The half-time substitute lobbed Dani Mallo less than 30 seconds after the restart as Rangers secured the double with a 1-0 Homecoming Scottish Cup final triumph.
Novo, who scored the winner when Rangers beat Hibernian to clinch the 2005 league title, has struggled to earn a starting place but he continues to pop up with crucial goals.
"Nacho is an incredible wee guy,'' said McCoist, who led Rangers throughout their cup campaign after Smith devolved responsibility to him.
"He's got a wonderful rapport with the fans and his team-mates. He deserves his success because he works so hard every day in training.
"The place lights up when the wee man walks in, whether it's a football park or a room.
"I know where he's coming from; it's not easy when you're coming off the bench. But I'm doubly delighted for him because he deserves it.
"Looking at the context of the game and the quality of the goal, it will take some beating.
"It's certainly one of the great goals to win any cup final in any country.''
Novo replaced Kris Boyd as Rangers matched Falkirk's 4-5-1 formation after being outplayed in the first half.
And McCoist admitted he could not have envisaged a pragmatic substitution having such a dramatic effect.
"We made a change because we felt we were losing out in midfield,'' McCoist said.
"To be fair to Barry (Ferguson) and Lee (McCulloch), they were outnumbered.
"We felt making the change we could put more pressure on their back four with Kenny (Miller) wide on the right and Nacho on the left.
"But it was beyond anybody's wildest dreams for a change like that to so quickly be beneficial. Because it was a goal worthy of winning any game of football.''
McCoist has enjoyed one of the greatest weeks in his career after Rangers stopped Celtic winning their fourth successive title.
"It just feels magic,'' the former Ibrox striker said.
"I thought last week was one of the best days of my football life but watching the players with the cup was fantastic.
"Falkirk deserve an awful amount of praise because they were terrific. It didn't surprise me; I said beforehand they can pass the ball as well as anyone in the league and they did that today. In the first half, they made it very, very difficult.''
Meanwhile, McCoist offered hope to Ferguson that he may be at the club next season.
After Pedro Mendes and Maurice Edu failed fitness tests, the 31-year-old made his first start for two months, since being suspended by the club following his part in the 'Boozegate' affair.
"He's got another year of his contract to go,'' McCoist said. "I'm hopeful that Barry will be here next year lifting the cup.
Nacho Novo admitted he did not even see his Scottish Cup winner fly into the net as Rangers completed a domestic double by beating Falkirk 1-0 at Hampden.
Novo, who has struggled to hold down a starting place at Ibrox, scored a stunning 30-yard volley just 29 seconds after his half-time introduction this afternoon.
"If I'm honest, I couldn't even see it. It passed so quickly,'' Novo told BBC Radio Scotland.
"I never saw it but I'm quite pleased for the goal. This was a really hard time for me.
"This will keep me going and looking forward to next season.''
Novo was impressed with the performance of Falkirk, who more than matched Rangers for the majority of the game.
The Spaniard said: "It's a team which always plays really good football and tries to play football, and a really good manager as well.
"They've been a really difficult side for us but I'm quite pleased for us as well to win it.''
Rangers' captain David Weir acknowledged Falkirk's first-half superiority.
"We were poor, they were very good,'' the defender surmised.
"What we were doing wasn't working so the manager made a decision and that's what he's good at.''
Of lifting the cup as skipper, Weir added: "It's a dream come true, it really is, and I'm going to enjoy every minute.''
Falkirk boss John Hughes was proud of his players.
He said: "All credit to them, they gave it a go. I think you know that with the fans staying behind.
"I'm so proud of them, I really am.''
He added: "My game plan was to keep the ball, ball-retention, taking our time.
"I said at half-time, we could've done it a lot quicker and we were giving them plenty of chance to get back behind the ball.
"I said at half-time, 'Make sure we keep it tight for 20 minutes'.
"And one minute into the second half, bang.''