Webber struck from the penalty spot in the 75th minute after midfielder Peter Murphy had been brought down by Charlie Taylor in the area.
Rotherham loan striker Odejayi made it safe, using his strength to hold off the Fleetwood defence and sliding the ball past keeper Scott Davies after 82 minutes.
But Fleetwood were counting the cost of missed chances earlier in the game.
Saturday's hat-trick hero Antoni Sarcevic blasted a shot from the edge of the area across the crossbar after 22 minutes from Gareth Evans' flick-on.
On-loan Blackpool striker Nathan Tyson also dragged an effort just wide for the visitors, while on the stroke of half-time Sarcevic's goalbound header was kept out by the home defence.
Stanley were limited to half chances, but Webber made no mistake from the spot and Odejayi pounced to make it eight league games unbeaten for one-time bottom club Accrington.