Rovers have carelessly dropped points against Sunderland, Manchester City and Bolton Wanderers since Allardyce replaced Paul Ince as manager in December. This was another occasion when poor finishing led to dropped points and Allardyce admits that more of the same will leave Blackburn in a perilous position come May.
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'This is two points dropped on the basis of our second-half performance,' Allardyce said. 'Jason Brown only had one real chance to save, but we had a number of good chances and didn't take any of them.
'We should be well out of trouble, but we're not, so we have to fight and battle and scrap to the bitter end. If we hold our nerve and use our experience, we should get out of the trouble we're in.
'It could be 38 points that keeps us up, but 40 is the magic mark now because the teams around us have been picking up three points. You can never tell, but we have to aim for 40 points.'
Comfortable in sixth position, Everton have Sunday's FA Cup quarter-final against Middlesbrough topping their agenda, so much so that Moyes was able to start with £15 million midfielder Marouane Fellaini on the bench.
Allardyce had no such luxury, but the 2-1 victory at Hull last Sunday had revitalised Rovers and their survival hopes. Avoiding relegation was a possibility rather than a probability when Allardyce inherited a sinking ship from Ince, but with a game in hand, against Fulham next week, the odds are now ever so slightly in Blackburn's favour.
Those prospects would have taken an early dent, though, had Everton forward Jo hit the target with an audacious attempt at goal inside the opening 20 seconds. The Brazilian, on loan from Manchester City, tried his luck from 25 yards and he was only denied the quickest Premier League goal of the season by a deflection that took the ball inches wide of the post and behind for a corner.
Such a bright start suggested that this would be a stirring derby, but it proved to be anything but in a tedious first half that proved utterly uneventful until Steven Pienaar brought a stunning save from Blackburn goalkeeper Brown, deputising for the injured Paul Robinson, with a right-foot strike from 12 yards after 40 minutes.
The half-time introduction of the 38-year-old playmaker Tugay, in place of groin-injury victim Vince Grella, at least gave Blackburn some much-needed impetus, but it did not last long. Roque Santa Cruz spurned a good chance created by Tugay and Stephen Warnock sent a free-kick against the crossbar before Jason Roberts shot wastefully at Tim Howard after being released by the Turkish veteran.
Everton barely threatened at the other end, however, save for an 88th-minute Joleon Lescott volley that flew the wrong side of the crossbar. It was one of those nights. Nobody was going to score if they continued beyond midnight.
Moyes said: 'It was a game of few chances, but nobody has a divine right to come to places like Blackburn and expect to get three points, so a draw isn't bad for us considering this is the first time that Blackburn haven't beaten us [in three attempts] this season.'