be convinced that ... = believe that ... 

在这个句式中，that从句是陈述语气，不是虚拟语气，时态受前面be的限制。

你提供的这个句子，needed 是实意动词，不是情态动词（后面跟了to do），是一般过去时，跟前面was 的时态一致。

再举数例：

I was convinced that he knew the truth. 我确信他知道事实。

They were convinced that he would not fail them.  他们相信他不会辜负他们的期望。

You will soon be convinced that she is right. 你不久就会相信她是对的。

You will be convinced that our product is superior. 你一定会相信我们的产品比别家的好。

I am not at all convinced that he's so cool a head. 我根本没法相信他的头脑这么冷静。

I am convinced that she is innocent. 我坚信她是清白无辜的。

I'm convinced that what you said is quite right. 我相信你说的是对的。

I was convinced that we were doing the right thing. 我坚信我们做的事是对的。

He was convinced that his theories were correct. 他确信他的学说是正确的。

They were convinced that we should give it a try. 他们认为我们应该试一下。

I felt convinced that he would succeed. 我确信他会成功。

I am convinced that is true. 我肯定那是正确的。