Here's the code:
$row = UserFields::where(['user_id' => $user['user_id'], 'field_id' => 'myfield'])->first();
$row->field_value = "new value";
For anyone having the same issue I managed to solve this by using basic lumen db query method:
app('db')->update("UPDATE `xf_user_field_value` SET `field_value` = ? WHERE (user_id = ? AND field_id = ?);", [ $newValue, $user_id, $column ]);
I have completely disabled Eloquent as it was overkill for me and not flexible enough for my needs.