A Carlos Tevez goal saw West Ham pull off an improbable 1-0 win at Manchester United in May 2007 to stay up by three points.
The current crop are bottom of the Barclays Premier League and two points adrift of safety with only four games remaining.
Great escape: Carlos Tevez celebrates scoring against Man United in 2007 as then-boss Alan Curbishley looks on
But Spector, who played in that win at Old Trafford, said: 'It's a very different team from the one we had that stayed up. It is a strong team right now, probably stronger than we had then.
'It has not happened for us this season but now we have to come together and get some points.'
Mark Noble will not play against Manchester City on Sunday after pulling stomach muscles against Chelsea on Saturday and the midfielder, 23, could yet miss the rest of the season amid fears he will need a hernia operation.
So near yet so far: Jonathan Spector goes close against Chelsea
Scott Parker is running in training after missing the past two games with an achilles injury, but manager Avram Grant will leave it late before deciding whether to play him against City.
Defender Spector, who is likely to deputise in midfield, added: 'They are massive games (coming up).
'There are quite a few players who could have a massive impact on these games that we have left.
'There have been times when we have played really good football and people have been saying, "They are too good to go down". They were saying that about Newcastle and we don't want to repeat what they did.'
West Ham have parted company with chief scout Roger Cross. Cross, 62, first joined the club's ground staff as a 15-year-old in 1964 and began his playing career at Upton Park.
Dog fight: Spector refuses to believe West ham are too good to go down
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