Invoke-SqlcmdUsingConnectionString

Invoke-SqlcmdUsingConnectionString.ps1

function Get-ConnectionStringComponent($components, $key) {
    $component = $components | where { $_.StartsWith("$key=") }
    $value = $component.Split("=")[1]
    return $value
}

function Invoke-SqlcmdUsingConnectionString($connectionString, $query){
    $components = $connectionString.Split(";")
    $serverInstance = Get-ConnectionStringComponent $components "Data Source"
    $database = Get-ConnectionStringComponent $components "Initial Catalog"
    Write-Host $dataSource
    Push-Location
    if($connectionString.Contains("Integrated Security=SSPI")) {
        $result = Invoke-Sqlcmd `
            -ServerInstance "$serverInstance" `
            -Database "$database" `
            -Query $query
    } else {
        $userId = Get-ConnectionStringComponent $components "User ID"
        $password = Get-ConnectionStringComponent $components "Password"
        $result = Invoke-Sqlcmd `
            -ServerInstance "$serverInstance" `
            -Database "$database" `
            -Username "$userId" `
            -Password "$password" `
            -Query $query
    }
    Pop-Location
    return $result
}

#$connectionString = "Data Source=..."
$result = Invoke-SqlcmdUsingConnectionString $connectionString "SELECT @@ServerName"
Write-Host $result.ItemArray

It won't cover all cases, but here it is.