City fought right to the end but ultimately came out unstuck against a better Buxton side on the day.
It was, however, City who got off to the better start and almost took t he lead two minutes in when a freekick curled just wide.
This was followed by a corner that saw Buxton go agonisingly close to heading the ball into their own net.
Glenn Steel then looked to cause problems with a long throw which found David Norris but his header went inches wide.
Lancaster continued to pile on the pressure but they just could not find that one clear cut chance to take the lead.
City didn’t capitalise though and were made to pay when Buxton won a penalty midway through the first half.
Bradley Grayson stepped up and buried it into the right corner, sending Jack Sims the wrong way.
City crumbled from there and found themselves 2-0 down when Tom Dean headed past Sims minutes later.
Lancaster responded with two back to back chances before the half time whistle but Buxton looked comfortable at the back and stayed strong.
It could not have been a worse start to the second half for City as Buxton added a third goal through Brad Grayson once again.
From there, Buxton took control and looked very comfortable on the ball.
City created a few more chances to get back into the game but it was too little too late once Ryan White knocked in a consolation goal in the 90th minute.