Blackpool had the ball in the net after 18 minutes but Brett Ormerod's header following a cross from Charlie Adam was ruled out for offside.
Varney was looking lively and skipped past Pantsil two minutes later, only to find the wrong side of the post with his effort.
Fulham responded and Etuhu saw his header from inside blocked after a corner by Damien Duff.
Fulham carved out a good chance in the 26th minute when Zamora worked his way into the area and tried to release Simon Davies.
But Alex Baptiste stood firm and made a crucial challenge just as his opponent was about to shoot.
Duff slipped the ball through to Moussa Dembele, who delivered it into the centre and Zamora sent a powerful header beyond goalkeeper Matt Gilks.
With the England squad about to be announced, it was good moment for Zamora to stress his aerial ability.
Fulham were keen to build on their lead and Zamora got goal-side of Stephen Crainey in the 49th minute but failed to put enough power on his shot and Gilks made the save.
Back came Blackpool and Ian Evatt's effort troubled Stockdale but he recovered with Ormerod handily placed to capitalise on any mistake.
Now it was Fulham's turn to come under some pressure and Stockdale spilled the ball following a challenge by Varney in the 61st minute.
It fell to Elliot Grandin and he found the net but again Blackpool were out of luck.
Referee Mike Oliver ruled that Stockdale had been fouled and disallowed Grandin's effort leaving manager Ian Holloway incensed on the touchline
Blackpool drew level in the 70th minute when Varney's shot came off Aaron Hughes and back to Pantsil, who lost his bearings and diverted the ball past his own goalkeeper.
With Zamora going off after taking a knock Fulham appeared to have lost their way a little.
It was new recruit Varney who took the plaudits with a glorious finish across Stockdale after a defence-splitting pass by Ormerod.
That goal only sparked Fulham into life, however and they drew level in the 87th minute when Etuhu sent a delicate lob over Gilks after being released by Dembele.
Holloway was delighted with the way his team responded after the 6-0 drubbing by Arsenal.
He said: ''It is a fantastic day for the club. What an achievement to get this game on. We have got to pinch ourselves we are at this level. I felt defensively we were much better today.
''Well done to everyone. Fulham are a fantastic side and Bobby Zamora looks a proper handful now. He is very clever and very difficult to mark. I was pretty happy when he went off. He looks a proper player.
''We scored two goals against Fulham. I watched Manchester United do that. If you had told me that after the Arsenal match I would have taken that.''
Fulham manager Mark Hughes thought they suffered when Zamora went to the dressing room but was with pleased with the team's character.
He said: ''We haven't picked up that many points on the road over the years. It is something we hope to improve on. It is key to whether we have a decent season.
''In the second half we didn't play as well as we normally can, made a couple of mistakes and let them get back into the game.
''It is good to see Dickson Etuhu getting on the end of something. He has not scored many goals in his career. We are trying to get him forward more. We lost a little bit of our focus when Bobby went off. That disrupted us somewhat. But we are pleased with an away point and will build on that.''