"That is the best experience I've ever had in club football. It's a pity we didn't win but hopefully we will next time," Lukaku said.
Sturridge started on the bench while midfielder Joe Allen made his first league start since March. Everton made just one change from the goalless draw at Crystal Pace before the international break, with Ross Barkley replacing Leon Osman.
Coutinho made the most of sloppy defending from Liverpool, finding himself unmarked at the far post from Steven Gerrard's corner to slot home his first goal of the season with a cool finish.
But Coutinho's goal did not silence the home fans for long as Mirallas leveled moments later with a shot high into the net after Liverpool defenders failed to clear a free kick in the box from Leighton Baines.
Lukaku then went close after being released by Pienaar but the in-form Mignolet denied his effort.
Everton was punished from another set piece, as Suarez put the Reds in the driving seat with a brilliant free kick bent around the Everton wall from 30 yards. It was Suarez's ninth league goal, a tally that makes him the joint top scorer with Sturridge.
Mirallas was lucky to escape a straight red car in the 33rd minute after an ugly foul on Suarez, with the Belgium forward stamping his studs on the striker's thigh. Suarez screamed in pain and lied on the pitch for several minutes but was able to return to the game.
Everton's hopes of turning the match around were dealt a blow at the start of the second half when Baines was forced to limp off the field with a foot injury in the 50th minute, being replaced by Gerard Deulofeu. The Spanish winger immediately threatened Mignolet but the Liverpool keeper proved decisive with a low save that kept the Reds in front.
Allen then wasted a good chance after being set up by Suarez, who had opened up the defense just in front of goal before releasing the ball for his teammate. But Allen slotted the ball wide from 10 meters, prompting an angry reaction from the Uruguay striker.
Everton kept pressing, forcing Mignolet to more saves and finally managed to level the score through Lukaku, who slid the ball through past his fellow Belgium teammate. The powerfully-built Lukaku then headed the ball into the back of the net before Sturridge rescued a point for the Reds following another pinpointed free kick from Gerrard.