The Chiefs led 10-0 at half-time after an opportunist try from scrum-half Haydn Thomas as Newcastle's habit of starting slowly and gifting tries struck again.
The Falcons were unable to come back from that deficit and although they did squeeze a losing bonus point out of the game with Tim Swinson's converted try two minutes from time, they remain well adrift at the foot of the standings
Tait admitted the "knives will be out" for him following the loss.
"We have to face up to reality and we have just had a meeting with the players and coaches about where we are," said the former Scotland and British Lions player who is in his second full season as Newcastle head coach.
Last year, Newcastle survived relegation only by points difference but this season they have been well adrift for much of the campaign.
Tait added: "It's going to be tough old battle now and we have to be realistic. I have to look at myself as well and I will do that and I will have a talk with the owner.
"I'll reflect on things and on my performances over the next few days. I had a dream and vision for this club and it hasn't quite worked and I have to speak to owner and see what happens. The knives will be out for me and quite rightly so.
"Maybe I'm writing myself off but it's hard not to do so after a result like that and things like that can cost you your job."
The win eases Exeter's relegation fears after they lost both of their holiday matches to London Irish and Harlequins.
They have now won three of their last four away encounters and will look to push on up the table.
"We set out to be ambitious but we were very well aware of the implications for both ourselves and Newcastle, " said Exeter head coach Rob Baxter.
"We knew they had said how important the game was to them and we knew we had to stop them coming out of the blocks quickly.
"We had to base things about getting a good start and hurt their ambitions and we did that. We aimed to keep the ball and to be the first on the attack and that went very well for us and Haydn Thomas scored a fantastic solo try.
"That got us on the front foot and then the game was a pretty even contest. We controlled large parts of the game without really breaking through and we weathered the storm in the second half and when we did break out, I think that was the game decided.
"It was little disappointing to kick the ball away with four minutes left and give them a try but all credit to Newcastle to attack for the losing bonus point and getting it. It was a really tough game, it's not easy to come here and get a result but I think we were deserved winners.
"It was a pivotal game for both clubs and there was a little bit of nervousness around. We seem to become little bit edgy when we feel there are points to be taken and if can free ourselves up to do that we'll climb the table."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.