Tyrone scored three goals past Down to progress to an Ulster SFC quarter-final against provincial holders Monaghan.
Mickey Harte’s team made Down pay for not killing them off in last week’s drawn game in Omagh, and were convincing winners in Saturday’s replay.
“We believed we could do better than we did last week,” said Tyrone boss Mickey Harte.
“Believing it and making it happen is a different story but I have to give full credit to the players for the effort they put in.
“Being in the next round with two games behind us is useful but again, Monaghan are waiting and they have had two chances to look at us and they will have learned quite a lot about us.”
Down boss James McCartan was proud of the effort put in by his team and was looking ahead to the qualifiers with some confidence.
“Tyrone got the draw last week and they came here on the front foot,” said McCartan.
“I don’t think the scoreline did our guys justice. I thought it was a wee bit cruel. We have a mix of experience and youth, and hopefully some of those young players will kick on.”