Javascript ES6, 44 bytes

$=_=>`$=(\^\_\^\)${($+'').slice(5)};$()`;$()

Output:

$=(^_^)=(\^\_\^\)${($+'').slice(5)};$()`;$()

Still working on golfing, but it works for now.

Vim, 33, 27 keystrokes

qqqqqS(<C-v>94_<C-v>94)q@q<esc>hh"qPq@q

Note that <C-v> means ctrl+v, and is byte 0x16 and <esc> is the escape character, and is byte 0x1B.

This just uses a slightly modified version of my Golf you a quine for great good! answer.

Outputs:

(^_^)S(^V94_^V94)q@q^[hh"qPq@q

This is valid since ^V is the way vim represents <C-v> and ^[ is the way vim represents <esc>.

The basic idea, is just to input the text (^_^) by its code points so we can avoid putting those characters in the source code. In insert mode, <C-v>number will insert the ASCII character of "number". However, since the challenge says:

The string (^_^) or any of its substring of length >=2 must not appear in initial source code, only in output.

This answer abuses the "substring" rule by only entering the codepoints of the ^ characters, and entering (, _, and ) directly.

Here's a gif that lets you see this solution in action, and puts the source code and output side by side for comparison:

CJam, 13 11 bytes

"(_)"
_p'^*

Online interpreter (-2 bytes thanks to @MartinEnder).

"(_)"             Push string
      _p          Duplicate and print repr with newline
        '^*       Join string with '^'

Python, 115 111 107 bytes

def f():s='def f():s=%r;print s%%s.replace(s[:5],"^".join("(_)"))';print s%s.replace(s[:5],"^".join("(_)"))

Call f() and the output is:

def f():s='(^_^)():s=%r;print s%%s.replace(s[:5],"^".join("(_)"))';print s%s.replace(s[:5],"^".join("(_)"))

Inspired in part by this answer to a similar question.

CJAM, 16 15 bytes

Try it here.

{"^(_)"(*}_~sss

Jelly, 22 bytes

“4094959441b³ỌØV”ṘVabc

Available at TryItOnline

Uses the built-in payload capable quine “ØV”ṘV
The abc on the end is just filler to be replaced
b³ converts the integer into base 100, resulting in [40,94,95,94,41]
Ọ casts to characters, resulting in (^_^)
So the whole result is “4094959441b³ỌØV”(^_^)

Go (golang), 131 bytes

This challenge must have an answer in Go!

package main;import"fmt";func main(){a:="package main;import\"fmt\";func(%c_%[1]c)(){a:=%q;fmt.Printf(a,94,a)}";fmt.Printf(a,94,a)}

Try it online!

C# 5.0, 715 bytes

I know, this is huge. Just wanted to add a C# solution.

/*^()_*/using System.CodeDom;namespace System{class P{static void Main(){var b="/*^()_*/using System.CodeDom;namespace System{{class P{{static void Main(){{var b={0};var f=new string(new[]{{b[3],b[2],b[5],b[2],b[4]}});var w=new IO.StringWriter();CodeDom.Compiler.CodeDomProvider.CreateProvider(\"CSharp\").GenerateCodeFromExpression(new CodePrimitiveExpression(b),w,null);Console.WriteLine(b.Replace(\"[4]}}}}}}}}\",f),w);Console.ReadKey();}}}}}}";var f=new string(new[]{b[3],b[2],b[5],b[2],b[4]});var w=new IO.StringWriter();CodeDom.Compiler.CodeDomProvider.CreateProvider("CSharp").GenerateCodeFromExpression(new CodePrimitiveExpression(b),w,null);Console.WriteLine(b.Replace("[4]}}}}",f),w);Console.ReadKey();}}}

The output is:

/*^()_*/using System.CodeDom;namespace System{class P{static void Main(){var b="/*^()_*/using System.CodeDom;namespace System{{class P{{static void Main(){{var b={0};var f=new string(new[]{{b[3],b[2],b[5],b[2],b[4]}});var w=new IO.StringWriter();CodeDom.Compiler.CodeDomProvider.CreateProvider(\"CSharp\").GenerateCodeFromExpression(new CodePrimitiveExpression(b),w,null);Console.WriteLine(b.Replace(\"[4]}}}}}}}}\",f),w);Console.ReadKey();}}}}}}";var f=new string(new[]{b[3],b[2],b[5],b[2],b[4]});var w=new IO.StringWriter();CodeDom.Compiler.CodeDomProvider.CreateProvider("CSharp").GenerateCodeFromExpression(new CodePrimitiveExpression(b),w,null);Console.WriteLine(b.Replace("(^_^)}}",f),w);Console.ReadKey();}}}

Available at Ideone.com

JavaScript (ES6), 91 bytes

There's already a JavaScript answer, but there's not a non-source-reading JS answer:

a="a=%s;co%s.log(a,uneval(a),`(${'^'}_${'^'})`)";console.log(a,uneval(a),`(${'^'}_${'^'})`)

This is based on my non-source-reading answer to Golf you a quine for great good!. Outputs

a="a=%s;co%s.log(a,uneval(a),`(${'^'}_${'^'})`)";co(^_^).log(a,uneval(a),`(${'^'}_${'^'})`)

This can easily be modified by moving around the second %s in the string. For example,

a="a=%s;console.log(a,uneval(a),`($%s_${'^'})`)";console.log(a,uneval(a),`(${'^'}_${'^'})`)

outputs

a="a=%s;co%s.log(a,uneval(a),`(${'^'}_${'^'})`)";console.log(a,uneval(a),`($(^_^)_${'^'})`)

MATL, 14 bytes

')_`_*i't&Dwqh

Produces the output ')_`_*i'(^_^)h.

Try it online!

Explanation

')_`_*i'  % Push this string
t&D       % Duplicate and get string representation (i.e. enclose with quotes)
w         % Swap
q         % Subtract 1. Transforms ')_`_*i' into the code points of '(^_^)h'
h         % Concatenate. Automatically casts code points to chars. Implicitly display