With record signing Andy Carroll having been unavailable all season due to injury, the Hammers' struggles for goals have seen them drop to 17th in the Premier League, just one point above Crystal Palace and Fulham.
But, in a Christmas message to supporters on the club's official website, Sullivan declared: "Families pull together during tough times - and these are tough times.
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"It's absolutely fantastic we've got into the semi-final of the [Capital One] Cup," he added, "and - who knows? Not only will we climb the league in 2014, we might have a cup final for everyone."
In March, the Hammers secured a deal to move into London’s Olympic Stadium from 2016 and Sullivan was keen to highlight the importance of the arrangement.
"Getting the Olympic Stadium will change the future of the club and increase our fanbase by hundreds of thousands of supporters," he enthused.
"It is one of the most important things we've done this year - and maybe in the history of the club - and it will do so much for the supporters.
"The Olympic Stadium will attract better players. If we're perceived as a big club with a great stadium in the heart of London, a lot of foreign players who might not have come to us - not even thought of it - will be attracted to West Ham.
"We're confident, in the summer, we can attract bigger players."