The Hammers endured a difficult start to the campaign as they battled with poor form and a plethora of injuries, but have responded well since the turn of the year.
The change in fortunes has helped propel the Londoners off the foot of the table and out of the relegation zone, although they still sit 17th in the table.
Rallying call: Grant is hoping his side can avoid the drop
West Ham midfielder Mark Noble argued earlier this week that the squad's talent belies their lowly position and under Grant's tutorship they can reach the Champions League next season.
The former Chelsea and Portsmouth manager laughed off Noble's comments but admitted the future is rosy if they can stave off relegation.
'It is good that players have ambition but first we need to stay in the league,' he said, speaking ahead of tomorrow's clash with Manchester United at Upton Park.
'As I said when I came here and even when we had difficult times, we have a vision at this club.
'Part of the vision is to be in the top third of the league and I think we can do it, but we need to do it step by step.
'First we need to stay in the league, second we need to analyse what we can do better in the next season.
'One thing I agree with Mark about - without mentioning the Champions League - is that this team is a good team and we can improve. In the next year we can get better and better.'
Sorely missed: Hitzlsperger (right) has recently returned to the Hammers side
The Hammers are in the midst of an impressive run that has seen them shore up their ailing defence and net more regularly.
The change in fortunes has coincided with Thomas Hitzlsperger's return from a thigh injury sustained in pre-season training.
The Germany international made his long-awaited Hammers debut in the 5-1 FA Cup defeat of Burnley and has since become an integral part of Grant's squad.
The former Chelsea and Portsmouth boss believes Hitzlsperger has been a catalyst to West Ham's recovery but refuses to think what could have been.
'My father always told me to look forward so to look back is only to learn from things.
'We didn't have Thomas and we know if he had been with us maybe we had, not for sure, more points.
'He is a very good player, a very good person and a top professional.
'You know we took him at the start of the season and he was our first signing because we saw that he would add a lot to the midfield.
'He is a player that knows how to do everything: defence, movement, assist and scoring goals.
'We missed him a lot and now we are playing like we did in the preparation games and his contribution you can see. It has been good and it can be better.'
Goal king: Demba Ba is proving a shrewd signing by Grant in his bid to beat the drop
This weekend Hitzlsperger and his West Ham team-mates face a tough test against leaders Manchester United.
The Hammers thrashed United 4-0 the last time the pair came to blows in the Carling Cup, but Grant knows they will be a vastly different proposition tomorrow.
'Manchester United are a very good team. They can beat any team on the day but I think also we can cause them a lot of problems.
'What happened in the Carling Cup was unique, you know, to beat Manchester United 4-0 in an official competition is something we will remember but the next game is different.'
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