Elgin then pushed for the opener. City talisman William Martin showed great desire and determinattion to beat two men on the right-hand side before slipping in Stephen Donnachie, who failed to apply the finishing touch.
Five minutes later Martin played in Stuart Cumming, whose weighted touch found Donnachie who cut the ball back for Andrew Roddie, but with the goal gaping the former Aberdeen man somehow shot wide.
Moments later Elgin had another great opportunity. Alan Dempsie again showed good vision, this time putting Donnachie clean through on goal but the forward failed to connect properly and the ball rolled into Paul Napier's hands.
Elgin were made to rue the missed opportunity as Peterhead scored from the resulting clearance. A mix-up in defence allowed Chris Hegarty to drift in from the left-hand side and roll the ball under Renton to the delight of the travelling support.
Peterhead almost scored within minutes of the restart when Scott Michie cut the ball back to Dougie Cameron, but his shot went wide.
Elgin nearly capitalised with Peterhead down to 10 men due to David Hagen recieving treatment. Roddie swung the ball in from the right, and it was met with Hugh Dickson, whose bullet header was bocked by Mathers. Minutes later Martin Bavidge had a great chance to double the Blue Toon advantage, but saw his effort land on the roof of the net.
Peterhead wrapped up all three points in the 88th minute with an incisive breakaway. Bavidge unselfishly led the ball back to Jamie Buchan, who stroked the ball past Renton from 30 yards. Elgin will be disappointed as third position is now beyond them.