Kevin Nolan scored both the goals as the Hammers chalked up a rare away victory, and maintain his own fine goal-scoring form - that's four goals in the last two games for the midfielder.
And Saturday's win sees Sam Allardyce's men move two points clear of the Premier League's bottom three.
Centre-back Collins said: "We just came here to play our game and get three points which is massive for West Ham. Villa tried to play their football but unfortunately it didn't work for them today.
"We were confident, we got a good draw at Chelsea and then we played very well at home last week. Losing Andy Carroll was a blow but Marco (Borriello) coming on at half-time changed the game really so we are delighted.
"We knew if we could get Kev scoring - he got two again today after getting two last week - he's massive for us and if he can carry on scoring between now and the end of the season, then I'm sure we'll be fine.
"We can (get ourselves out of trouble). It sounds funny but we were always confident. You try not to make excuses about injuries and decisions and stuff but they were going against us and now we seem to have turned a corner.
"We've had three massive results on the bounce which has turned around our season."
Asked if his former club Villa can avoid the drop too, he added: "I'm sure they can. They are a massive club and they've got some big players - obviously the boys up front, if they can score the goals then there home form is going to be big.
"I'm sure they can turn it around and hopefully they can stay up."