The stalemate means the Three Lions have only won one of their last 15 Test matches as the captain's 24th Test century and bold declaration proved in vain.
The tourists had set West Indies a target of 286 in 71 overs and had their hosts wobbling on 67-4 before Nkrumah Bonner and Jason Holder dug in.
A flat pitch and an injury to Mark Wood compounded England's frustration, as Bonner and Holder dropped anchor for 239 deliveries and an inevitable draw.
"We want to win Test matches. It's been a lean year, a lean tour of Australia and this was a chance to get back on track, to go for it," Root told reporters.
"I'm really proud of this team. I thought the attitude throughout the whole week was just fantastic. It's filled me with a huge amount of confidence. We threw everything into the game and to be in the position we were is a really pleasing performance on what turned out to be a very docile wicket.
"They never stopped believing and that's what we want to encourage in this team."
Root and his England team-mates are trying to restore confidence off the back of a torrid winter in which they were thumped 4-0 by Australia and an all-too-familiar outcome looked inevitable following a collapse on day one. Zak Crawley (121) and Root (109) gave England hope in the second innings and the captain said there were many positives to take.
He said: "We want to keep believing, keep fighting and show the supporters that have come out to watch here that we're desperate to get some wins under our belt.
"I really hope the guys gain so much confidence from this. We're very buoyant, we created pressure in the field. Over time stuff like that can make a real difference to a team."
The two sides meet again for the second Test at the Kensington Oval in Bridgetown on Wednesday 16 March (14:00 GMT).