The tourists were in the box seat for much of the Boxing Day Test, bowling out Australia for a below-par 327 on an unresponsive MCG pitch and then taking a 164-run innings lead on the back of Alastair Cook's herculean unbeaten 244.
It was only the mastery of Steve Smith who kept England at bay, batting out the entire last day with an unbeaten 102 for a draw to leave the tourists still without a win in four Tests.
Root won't countenance anything but victory in Sydney to finish the Test series, bedevilled by off-field troubles.
"Disappointed that we couldn't quite crack it open today but we tried absolutely everything and the efforts and work- rates from all the guys was there all day," Root said.
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"I am very proud about the way we went about it. Off three very difficult games and to put in a performance like that is really pleasing and that's what we're about as a side.
"Cookie to go on and make a massive double-hundred like he did was exactly what you're after following a strong two days with the ball."
Coming after the crushing disappointment of relinquishing the Ashes in a dispiriting innings defeat in the third Perth Test, Root said his team had showed its character in Melbourne.
"In terms of our squad the character is never something that I question and the way that we apply ourselves and the attitude that we take is always there," he said.
"That's the response you want to see from your players," he added.
"We are a better team than what we've played on this trip and that is a benchmark for our side moving forward and try to make sure that we go one better in Sydney and get that win that we want.