Coventry City dominated large spells of possession without ever truly threatening, and it was Hull who struck on the counter, Millar finishing clinically after being released into space during a rapid break. The result puts Hull City two wins from two, while Coventry remain without a point.
Frank Lampard was left to rue his side's inability to convert territorial dominance into chances.
"For a game we controlled a lot of the possession we were too slow with it. Against a back five and a team who want to slow you down," he said. "We created more problems than Hull did. But it wasn't like us. Normally we're more aggressive. There's some serious lessons to learn.
"Quality as well - you have to work towards quality. It was too much slow play from us. It's hard against a back five. When you cross they have players to deal with it."
Lampard also pointed to the manner of the winning goal as a source of frustration.
"The counter-attack goal was our corner - you have to stop it by any means. Those details are crucial," he said, before reflecting on the wider challenge of adapting to the top flight.
"The Premier League brings it with feeling around it, it's a step up. It's a mental side for the players - I want them to get over it. They have to get to grips with it. We have to get there quickly. You talk about experience but it's moments on the pitch."
Jakirović was satisfied with how his side managed the game after a difficult opening period.
"The team played a good game. They tried to overload our right side. Some changes at half-time, we had some good counter-attacks. We needed better decisions," he said. "Second half, a lot of crosses came into our box and we controlled everything there. In the end I'm very pleased. Two games, six points - not bad."
The Hull boss explained his half-time reasoning, with fitness concerns shaping his substitutions.
"At half-time we had problems with fitness and injury. We knew they would put more and more risk and there would be big space behind the back and we used that," he said. On midfielder Lucas Gourna-Douath, introduced from the bench, Jakirović added: "He also struggled with some moments because he didn't pass all of pre-season. He played good. He's a good young player and we can improve him. It's great experience for him. We will grow every game."