Cook, who scored a meticulous second innings century as England won the first test at Lord's, welcomed the appointment of Australian Bayliss ahead of the Ashes test series against his compatriots.
"Andrew (Strauss, England's director of cricket) obviously decided to get the best man for the job," Cook told reporters on Thursday. "To coach England is a huge role and Trevor's experienced in all forms of the game and, everywhere he's gone, he's been successful."
Reflecting on England's entertaining first test triumph, he said: "It was a 'fantastic' week at Lord's. It was a great test match to watch and both sides played really well. We get great support here at Headingley and I'm glad there's a rush for tickets.
"It was a rocky couple of weeks for England, off the field, and we're still in limbo until Trevor turns up. The way Paul (Farbrace, interim coach) has taken the side, he's done a fantastic job.
"The challenge for us, (as) players now, is can we back up a great performance in the last three days of that test match and try to win the series..."
Looking further ahead to Bayliss's arrival next month, Cook added: "His one day record is very good as well so it'll be very interesting to hear his ideas on that side.
"I don't really know him that well, but I played against his Sri Lankan team.
"Everyone I've spoken to about him has spoken very highly of him. It's a great coup to have him and the guys are looking forward to him turning up."
(Writing by Timothy Collings; editing by Justin Palmer)
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