The promoted visitors took a surprise lead in the 14th minute of the Barclays Premier League opening-day clash at Goodison Park through Miguel Layun, and the Toffees struggled to muster much of a response before the interval.
They improved in the second half and broke the Hornets' resilience to equalise through a sweet Ross Barkley strike in the 76th minute.
Watford then regained their advantage thanks to substitute Odion Ighalo in the 83rd minute, but Everton levelled again three minutes later thanks to another player called off the bench, Arouna Kone.
Reflecting afterwards on his side's display, Martinez said: "I thought we saw two very different performances from us.
"I think in the first half, we were clearly not at the level, speed, tempo and intensity that Watford had.
"Without knowing much about how these players gelled together (Watford had six debutants in their first XI), what we knew was that we were facing a newly-promoted side - that means you bring enthusiasm, bravery and an incredible intensity.
"But I think you saw in the first half a team that was in pre-season mode
If we are like that, we cannot compete in this league, and after they scored, we had an incredible mountain to climb.
"The second half was completely different and I think the attitude changed completely.
"We got in that competitive mode
I think the game changed a lot with the introduction of Arouna Kone and, with a couple of tactical switches, everything became a lot clearer for us.
"We became a little bit freer with what we did in front of goal, and you cannot underestimate coming back from behind twice in the game - that pleases me."
Martinez - eager to see his side improve on last season's disappointing 11th-placed finish - had plenty of praise for playmaker Barkley, Everton's outstanding performer.
The manager said: "Highlighting Ross - he has scored a goal that only he can score, with that technique and class but also with the maturity of a new, refreshed Ross Barkley.
"Two years ago he scored a similar goal in the opening game of the season, and that was a brave, raw, enthusiastic Ross Barkley.
"I think today we saw a very composed Ross Barkley
I am very excited after that performance from him."
As well as 10 new players, Watford have also acquired a new boss over the summer following their promotion from the Sky Bet Championship - Martinez's compatriot Quique Sanchez Flores.
And regarding his first competitive assignment in the job, Flores said after the match: "Having conceded the second goal so late in the game, the first feeling when you go to the dressing room is that you have lost two points.
"But we have played against an amazing team
Everton were playing well - they had had to change their way of playing.
"If we are honest, we have to recognise that this result is okay."
Asked about his confidence of avoiding relegation this season, Flores said: "I am really confident in this group
We have been working very hard over the last six weeks to be competitive.
"Of course, we can improve
But for this first match, I am happy with the performance."
Flores also said Watford were close to securing yet another signing - Guangzhou Evergrande's former West Ham midfielder Alessandro Diamanti, who played for Fiorentina on loan last season.
Source : PA
Source: PA