Esajas struck with an eye catching 30-yard volley after 19 minutes before Jamaican international Johnson drilled in a left foot strike before grabbing a second on 50 minutes.
Wednesday, looking for a good cup run after falling at the first hurdle against lowly neighbours Rotherham last season in a penalty shoot out, carved out the best early chance.
Johnson cut inside to pick out Marcus Tudgay 12 yards out but he was unable to control the ball as keeper Kenny Arthur smothered his shot.
The Sheffield side were handed an ever better opportunity minutes later with Johnson again the provider. His low cross from the left flank fell nicely for Tommy Miller, making his debut after a free transfer move from Ipswich, the lanky midfielder drilling in a low strike stopped by the outstretched leg of the Rochdale keeper.
Rochdale finally caved in as Wednesday stormed ahead with a spectacular strike. Tudgay challenged as the ball fell to winger Esajas who finished with a dipping volley with Arthur helpless.
Wednesday struck again two minutes later with the Lancashire side still shell-shocked, the impressive Johnson shrugging off two challenges as he picked his spot before slamming the ball into the left hand corner.
Rochdale, who lost striker Adam Le Fondre to Rotherham in a £100,000 deal earlier in the day, almost found a way through minutes before the break. Chris Dagnall saw his stinging shot from 12 yards kept out by Lee Grant as Wednesday eased off.
Johnson grabbed his second of the night on 50 minutes. Centre back Darren Purse punted the ball forward and Johnson cooly hooked the ball over the advancing Arthur from the edge of the area.