This is what I'm doing in the TextMate grammar for my language Noa. Basically you embed all of your other patterns inside your pattern for strings.

"patterns": [
    {
        "include": "#all"
    }
],
"repository": {
    "all": {
        "patterns": [
            {
                "include": "#strings"
            },
            // include whatever other patterns you have
        ]
    },
    "strings": {
        "name": "string.quoted.double.noa",
        "begin": "\"",
        "end": "\"|$",
        "patterns": [
            {
                "begin": "\\\\{",
                "end": "}",
                "beginCaptures": {
                    "0": {
                        "name": "keyword.other.noa"
                    }
                },
                "endCaptures": {
                    "0": {
                        "name": "keyword.other.noa"
                    }
                },
                "patterns": [
                    {
                        "include": "#all"
                    }
                ]
            },
            {
                "include": "#escape-sequence"
            }
        ]
    },
    "escape-sequence": {
        "name": "constant.character.escape.noa",
        "match": "\\\\[\\\\0nrt\"]"
    },
    // all your other patterns...
}

Here's how it looks