Hearts manager Robbie Neilson focused on the positives – including the impact of Jamie Walker – after failure to add to a 1-0 lead over Inverness consigned the Jambos to a Premier Sports Cup trip to Celtic Park.
Hearts needed to win and score at least twice to secure a place among the second-round seeds but did not make the breakthrough until substitute Walker's 75th-minute goal.
Neilson's side scored eight goals without reply in four Group A victories but finished behind fellow group winners St Mirren and Dundee United on goals scored.
Neilson looked forward to their trip to Celtic Park, which will come two weeks after hosting the Hoops at Tynecastle on their top-flight return.
"We created loads of chances and were sound defensively," the Hearts boss said on his club's official website. "We just didn't have that final moment.
"We won all our games and haven't conceded a goal. Yes we'd have liked to have scored loads of goals but teams make it difficult and sit in.
"We have three massive games now with St Mirren in between the two Celtic matches. It's a huge start to the season and we're really looking forward to it."
Walker has not started a game since Hearts' Scottish Cup defeat by Brora Rangers in March but he was the key player.
Hearts forwards Liam Boyce and Armand Gnanduillet had half-chances in the first half while Alex Cochrane pulled off a brilliant block from Tom Walsh's shot at the other end.
The chance of the first half came in the final minute when former Hearts goalkeeper Mark Ridgers made an excellent stop from Finlay Pollock to stop the 17-year-old converting Josh Ginnelly's low cross.
Ridgers saved from Gary Mackay-Steven before Gnanduillet and Pollock both hit the post in quick succession, and the Ivory Coast frontman was denied by the goalkeeper with time beginning to run out.
Walker came off the bench and lit up the game, taking a Boyce pass in his stride and firing a left-footed drive home from 18 yards in the 75th minute.
The midfielder forced three saves from Ridgers as Hearts pushed for the crucial second and was hacked down by Michael Gardyne in a foul which sparked a straight red card with two minutes left.
Mackay-Steven flashed a shot over on the half volley in the fifth minute of stoppage-time as Hearts just missed out on a seeded spot.
Neilson said: "I thought Jamie was great when he came on. The game started to open up and he gets into pockets of space.
"He is probably one of the best we've got when it's like that – taking the ball and driving at defences. He just needs to work on his fitness so he can do that for 90 minutes.
"To his credit, he is working extremely hard to be as fit as he can be."
