The Australia international penned a one-year deal, with the option of another 12 months in the club's favour, at The Hawthorns yesterday.
Rogic, 29, is a favourite at previous club Celtic where he played almost 300 games in nine years but departed Parkhead a year before the end of his contract north of the border this summer.
Bruce revealed Rogic, who is unlikely to feature against Blues tomorrow, was thought to have a move lined up earlier in the summer. He had been working on fitness in his homeland but arrives in the Black Country 'three months behind' his team-mates' match sharpness.
"He said something along the lines of he was four miles from Sydney and then he arrived on Friday, we had a conversation on Monday and Tuesday and it was done by Saturday," Bruce said on initial talks with midfielder Rogic.
"It was really quick and I'm delighted, we've just got to get him up to speed now because our lads are three months ahead of him, which is a big call.
"There's the only drawback, but he'll do his best, he's a good player."
Bruce added: "Earlier in the window we thought he was done and dusted somewhere else, that didn't happen, didn't materialise.
"I haven't really asked Tom what the situation was, but we were under the assumption that he was gone somewhere else.
"He came on the radar very quickly, it got done very quickly, considering he was in Australia. He couldn't wait to get here. It was one of the easiest ones we've done, and we've got a very good player on our hands.
"He's only 29, played 300 times for Celtic, I've seen him many times, we're pleased that we got that one done."
Albion's midfield has mostly consisted of a partnership between Jayson Molumby and Okay Yokuslu in recent weeks, with John Swift playing a more advanced role.
Bruce admitted Rogic's arrival could provide the platform for a formation tweak in midfield as he looks to inject quality into his side.
"The more good players you have the better," he said. "We're always greedy. Can we play the two of them? Can we play a six and two eights? There's a possibility.
"It's important to be flexible. He's a very good footballer and one area we've tried to get better is at the top end of the pitch. We haven't quite got the balance right yet but I believe we're close. We've seen bits, but defensively is my biggest concern. We've given away poor goals."
Bruce added of his most recent foray into the free-agent market: "Well done to everyone concerned. We've worked tirelessly – it's not an easy market when you can't spend – and we've had to make big decisions in letting people out.
"Some people might've looked at some and thought 'I don't know what to make of that' but we had to make those decisions. It won't take one window, but we've made good strides. That's seven or eight, they've got to be judged on what they can do."