CSS: Input fields with submit buttons styling - css

Question

I am currently working with an input field that will do search for my page. But I am undergoing some css styling issues. I have placed inside the input field the submit button which is an image of a magnifying glass. the problem is that the button does not stay in its place through cross browsers. It looks ok in firefox but others browsers it looks bad.
How can i get the submit button to stay inside the input field at all times? EXAMPLE
thank you!
CSS related to object
<style>
.search-input {
padding: 0 5px 0 22px;
border: 2px solid #DADADA;
height: 30px;
font-size: 12px;
line-height: 30px;
border-radius: 25px;
-moz-border-radius: 25px;
-webkit-border-radius: 25px;
background: #FFF; /* old browsers */
}
.search-input li {
list-style: none outside none;
}
.search-submit {
width: 13px;
height: 13px;
border: none;
background: url(http://webprolearner2346.zxq.net/css-test2/images/mag-glass.png) no-repeat;
display: block;
position: absolute;
right: 170px;
text-indent: -9999em;
top: 60px;
}
.search{
float: right;
margin-right: 30px;
margin-top: 35px;
}
</style>
HTML
<div id="header">
<div class="search"><input type="text" class="search-input" name="search" value="Search"/><input type="submit" class="search-submit" /></div>
</div>

Answer 1

Use this much better approach I came across. It works well and tested with web developer tools. Cheers#!!!
HTML
<div id="header">
<form id="search">
<input id="searchField" type="text" />
<input id="searchSubmit" type="submit" value="" />
</form>
</div>
CSS
#search{
float: right;
margin-right: 30px;
margin-top: 35px;
}
#searchField{
border: 1px solid #FFEDE8;
float: left;
height: 20px;
padding-right: 25px;
width: 140px;}
#searchSubmit{
background: url("http://webprolearner2346.zxq.net/css-test2/images/mag-glass.png") no-repeat scroll 0 0 transparent;
border: medium none;
cursor: pointer;
height: 20px;
margin-left: -22px;
margin-top: 5px;
width: 20px;
}