Cook said he had every faith that Root, Compton, along with Jonny Bairstow, can prosper as England seek to regain their world number one test ranking despite failing to convince in the drawn series in New Zealand, and for much of the opening test against the same opposition at Lord's last week.
Cook acknowledged, ahead of the second and final test against the Black caps at Headingley starting on Friday, that England, still missing the injured Kevin Pietersen, had not been piling up the runs compared to past teams.
"Both sides found it quite tough to bat on that wicket (at Lord's)," he told a news conference on Thursday.
"Clearly our batting line-up has changed from a couple of years ago - there was a lot of experience then with Colly (Paul Collingwood) and Straussy (Andrew Strauss). We were obviously getting bigger scores than we have done recently.
"There has been a slight changing of the guard - young guys coming in, taken their time to find their feet at international level but they are quality players and I have no doubt in their ability."
Opener Compton scored centuries in the first two tests in New Zealand but has made just 46 runs in his last four test innings.
Bairstow, filling in for Pietersen who is expected back for the Ashes series against Australia starting in July, is averaging just over 25 in seven tests while Yorkshire team mate Joe Root, a year younger at 22, has impressed with his strokeplay and maturity at the crease.
Root top-scored in the Lord's test with 71 and there has been talk of him partnering Cook at the top of the order.
Cook said that prospect was for the future, with Compton, a late entrant to the test arena at 29 having been given the opportunity following Strauss's retirement, fully deserving of his place as opener alongside the captain.
"I'm sure down the line you will see Joe Root opening the batting. But Compo has got the shirt at the moment and fully justifies that selection," the skipper said.
"He has shown us the determination and the talent he has got at the top of the order and the character you need to fight there.
"Rooty has always opened the batting (in county cricket) - he sees himself as an opener and he's been given his chance (with England) in a different position."
Pietersen, who will not play in next month's Champions Trophy after suffering a knee problem in New Zealand, is an automatic choice to come straight back into the side when fully fit.
"He's (Pietersen) progressing well from his injury... chatting to him last week at Lord's he sounded positive, so fingers crossed," Cook said.
"We all know, when hopefully Kevin comes back and is fit, his record and class demands that he plays for England. That creates competition for places and the guys in the changing room will be desperate to score runs."
England, Cook said, were fully focused on completing a series win over New Zealand and maintaining the momentum from their crushing 170-run first test win at Lords when they skittled the tourists for just 68.
Mentioning the Ashes "had not been banned" in the dressing room, Cook said in batting down talk that coach Andy Flower had ordered silence on the matter.
"It's not been banned but it's important we stay in the present," he said.
"We know, what's coming up in the summer, how exciting it is to play in an Ashes series. But ff you start thinking what could happen you can take your eye off the ball."