Demba Ba's double cancelled out earlier strikes from Leighton Baines and Victor Anichebe.
The match descended into controversy afterwards, where firstly Marouane Fellaini saw a goal incorrectly chalked off for offside before Anichebe’s close-range header was deemed not to have crossed the line by the officials, despite replays showing otherwise.
But Taylor insisted their second-half spirit deserved such good fortune despite admitting the Magpies were lucky in those instances.
He told Newcastle’s official website: "We've had a look at it and for me it does look like it was past the line so we got lucky.
"But sometimes you need that. For Newcastle United, a few years ago we wouldn't have got that.
|MOYES LEFT FRUSTRATED
|7/1||Newcastle are 7/1 with Bet365 to beat Maritimo 1-0
"I think we deserved to maybe even win the game - [Hatem] Ben Arfa got a good chance as well - and the lads fully deserved it. We worked so hard."
The centre-back acknowledged his side’s poor showing in the first half but pointed out that Everton are a notoriously difficult side to beat, especially at Goodison Park.
He added: "The first half performance was poor, we came in and I think we were lucky to be 1-0 down but second half we just got at them and we caused them a lot of problems.
"This is a hard place to come. Every time I've came here it's always been a hard game. I remember a few years ago when we were getting beat 2-0, then me and Damien Duff scored but that's what it's like here - it's so difficult.
"The ground feels dead tight and they've got great supporters here. They're very well organised and they're hard to beat."
Demba Ba was left out of the starting line-up but came on at half-time, striking twice to deny Everton and Taylor was effusive in his praise for the Senegalese front man.
"It would've been easy for him just to sulk about not starting but he came on and turned the game on its head for us," Taylor continued.
"He was fantastic when he came on, caused problems and put himself about. He's a goal machine - even when you play against him in training, he's a strong fella. You give him half a yard and he'll score."