East Fife assistant Dougie Anderson admitted the Methil men fell flat after last weekend’s win in Stirling.
Second-half goals from Thomas Reilly and Kyle McLeod were enough to ensure the points left with the visitors.
Anderson said: “Last week we looked a real threat the whole game against Stirling, but we really lacked a threat against Elgin.
“In the first half we had the wind and wanted to put Elgin under pressure but didn’t really do that.
“We really needed to go in at half time with the lead. We huffed and puffed but just didn’t create anything, and our set-plays weren’t as good as they could be.”
He added: “We couldn’t get ‘Fash’ (Nathan Austin) into the game.
“Kyle Wilkie worked hard but couldn’t find that final pass.
“Against the wind we kept the ball a bit better.”
A strong, swirling wind made conditions almost impossible for both sets of players. Any high ball was soon whipped back, while even those along the floor were being swept out of the park.
East Fife had the wind at their backs in the opening half but failed to capitalise on it.
Nathan Austin hit the side netting, a long-range Ross Brown strike was saved, and Stevie Campbell cleared the bar in a half short of incident.
Elgin took an early lead in the second half when a slick move was finished off in style by Reilly.
The home side went on to have the bulk of play in the second half with Gary Naysmith introducing Jordan Austin and Jamie Insall.
But despite the attacking intent, the Fifers were unable to find a way past Mark Hurst in the Elgin goal.
They were sucker-punched late on when Jim Weir’s side grabbed the goal which ensured the win to send them top of League Two. Marc McKenzie sent in a cross the Fife defence failed to deal with, and the loose ball was netted by McLeod.