diff --git a/docs/rules/expect-expect.md b/docs/rules/expect-expect.md
index 6ba708903..258114cad 100644
--- a/docs/rules/expect-expect.md
+++ b/docs/rules/expect-expect.md
@@ -75,3 +75,24 @@ test('returns sum', () =>
     .run();
 );
 ```
+
+Examples of **correct** code for working with the HTTP assertions library
+[SuperTest](https://www.npmjs.com/package/supertest) with the
+`{ "assertFunctionNames": ["expect", "request.get.expect"] }` option:
+
+```js
+/* eslint jest/expect-expect: ["error", { "assertFunctionNames": ["expect", "request.get.expect"] }] */
+const request = require('supertest');
+const express = require('express');
+
+const app = express();
+
+describe('GET /user', function() {
+  it('responds with json', function(done) {
+    request(app)
+      .get('/user')
+      .expect('Content-Type', /json/)
+      .expect(200, done);
+  });
+});
+```
diff --git a/src/rules/__tests__/expect-expect.test.ts b/src/rules/__tests__/expect-expect.test.ts
index 3ef7ed9a8..209a48505 100644
--- a/src/rules/__tests__/expect-expect.test.ts
+++ b/src/rules/__tests__/expect-expect.test.ts
@@ -26,6 +26,52 @@ ruleTester.run('expect-expect', rule, {
       code: 'it("should return undefined",() => expectSaga(mySaga).returns());',
       options: [{ assertFunctionNames: ['expectSaga'] }],
     },
+    {
+      code: `test('verifies expect method call', () => {
+        class Foo {
+          expect(k) {
+            return k;
+          }
+        }
+        new Foo().expect(123);
+      });`,
+      options: [{ assertFunctionNames: ['Foo.expect'] }],
+    },
+    {
+      code: `test('verifies deep expect method call', () => {
+              class Foo {
+                expect(k) {
+                  return k;
+                }
+              }
+              let tester = {
+                  foo: function() {
+                      return new Foo()
+                  }
+              }
+              tester.foo().expect(123);
+            });`,
+      options: [{ assertFunctionNames: ['tester.foo.expect'] }],
+    },
+    {
+      code: `test('verifies recursive expect method call', () => {
+                    class Foo {
+                      expect(k) {
+                        return this;
+                      }
+                      bar() {
+                          return this;
+                      }
+                    }
+                    let tester = {
+                        foo: function() {
+                            return new Foo()
+                        }
+                    }
+                    tester.foo().bar().expect(456);
+                  });`,
+      options: [{ assertFunctionNames: ['tester.foo.bar.expect'] }],
+    },
     {
       code: [
         'test("verifies the function call", () => {',
diff --git a/src/rules/__tests__/no-jasmine-globals.test.ts b/src/rules/__tests__/no-jasmine-globals.test.ts
index 254c62d30..abfc0aa4e 100644
--- a/src/rules/__tests__/no-jasmine-globals.test.ts
+++ b/src/rules/__tests__/no-jasmine-globals.test.ts
@@ -13,6 +13,7 @@ ruleTester.run('no-jasmine-globals', rule, {
     'test("foo", function () {})',
     'foo()',
     `require('foo')('bar')`,
+    '(function(){})()',
     'function callback(fail) { fail() }',
     'var spyOn = require("actions"); spyOn("foo")',
     'function callback(pending) { pending() }',
diff --git a/src/rules/utils.ts b/src/rules/utils.ts
index 9cc2e7a03..c5b0a6eee 100644
--- a/src/rules/utils.ts
+++ b/src/rules/utils.ts
@@ -614,6 +614,10 @@ export function getNodeName(node: TSESTree.Node): string | null {
       break;
     case AST_NODE_TYPES.MemberExpression:
       return joinNames(getNodeName(node.object), getNodeName(node.property));
+    case AST_NODE_TYPES.NewExpression:
+      return getNodeName(node.callee);
+    case AST_NODE_TYPES.CallExpression:
+      return getNodeName(node.callee);
   }
 
   return null;