Victory left Everton three points behind city rivals Liverpool, who currently occupy fourth place and with it the final Champions League spot on offer to English clubs, with both Merseyside teams having five domestic league matches to play before the end of the season.
For basement club Derby their 24th defeat from 33 league games this term left them on 11 points - still four shy of the Premier League lowest total of 15 for a single season set by Sunderland in 2005-06.
Everton nearly went ahead in the first minute when Derby defender Dean Leacock headed towards his own goal only for the ball to bounce down into the turf and over the crossbar.
Derby though, with hailstone falling, did create some chances of of their own and it needed a smart save from Everton’s United States international goalkeeper Tim Howard to turn away an angled shot from Mike Sterjovski.
The only goal of the game came in the 56th minute when midfielder Osman, allowed time and space to run onto Manuel Fernandes’s through-ball, shot powerfully past former Manchester United goalkeeper Roy Carroll.
Results:
Played on Sunday
Everton 1 Derby County 0
Middlesbrough 2 Manchester United 2
Played on Saturday
Arsenal 1 Liverpool 1
Aston Villa 4 Bolton Wanderers 0
Blackburn Rovers 1 Tottenham Hotspur 1
Fulham 1 Sunderland 3
Manchester City 0 Chelsea 2
Newcastle United 3 Reading 0
Wigan Athletic 2 Birmingham City 0
—Agencies