The Egypt international had his medical in London and has put pen to paper on a four-year contract, with the Hoops having the option to extend by a further 12 months.
The clubs have agreed an initial fee of around £6m, with £3m in add-ons.
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A significant chunk of the bonus payments will be paid out if Celtic reach the league phase of the Champions League later this month.
There is now time for Hassan to be registered in time to be part of the squad for Celtic's Champions League play-off against LASK Linz next week. The deadline to register players is Thursday, with the opportunity to add two more before Tuesday.
Hassan 'grateful' to be joining 'biggest club in Scotland'
Speaking to club media, he said: "I'm really grateful to be here and really happy.
"Celtic is a massive club. When I was growing up, the club was always in the Champions League and had big players. I'm really looking forward to getting on the pitch with the guys.
"At 24, I think it's a good moment for me to move away from Spain. I've been there for six years.
"With Celtic's history, all the great players, the fans and the atmosphere, I think this is the perfect moment for me.
"I know I'm joining a big club, the biggest club in Scotland, and I'm really looking forward to taking the next step."
Hassan, who impressed at this summer's World Cup, had been a long-term target for the Hoops and they were in the market for a left-footed right-winger in this window.
He made 38 appearances for Real Oviedo last season, contributing three assists from his preferred position on the right wing.
He is Celtic's third summer signing following the arrival of strikers Camilo Duran and Kasper Hogh.
Reflecting on Hassan's signing, Celtic manager Martin O'Neill said: "We are delighted to bring Haissem to Celtic and I think the club has done really well to secure a player of such quality, with a number of other clubs interested in him.
"He is someone who I think has the ability and experience to do very well for us, performing at a high level in domestic and international football.
"I know he is excited about being with us and I am sure our fans will enjoy the energy and attacking intent which Haissem will bring to matches. I hope he will become a very important member of our squad going forward."
