As of npm cli v8.3.0 (2021-12-09) this can be solved using the overrides field of package.json. As described in StriplingWarrior's answer
For example, the project has typescript version 4.6.2 as direct development dependency and awesome-typescript-loader that uses old version 2.7 of typescript. Here is how you can tell npm to use version 4.6.2 of typescript for awesome-typescript-loader:
{
  "name": "myproject",
  "version": "0.0.0",
  "scripts": ...
  "dependencies": ...
  "devDependencies": {
    "typescript": "~4.6.2",
    "awesome-typescript-loader": "^5.2.1",
    ...
  },
  "overrides": {
    "awesome-typescript-loader": {
      "typescript": "$typescript"
    }
  }
}
If you don't use typescript as direct development dependency, then you have to write 4.6.2 instead of $typescript in overrides section:
{
  "name": "myproject",
  "version": "0.0.0",
  "scripts": ...
  "dependencies": ...
  "devDependencies": {
    "awesome-typescript-loader": "^5.2.1",
    ...
  },
  "overrides": {
    "awesome-typescript-loader": {
      "typescript": "~4.6.2"
    }
  }
}
For using the latest version of dependency:
{
  "name": "myproject",
  "version": "0.0.0",
  "scripts": ...
  "dependencies": ...
  "devDependencies": {
    "awesome-typescript-loader": "^5.2.1",
    ...
  },
  "overrides": {
    "awesome-typescript-loader": {
      "typescript": "latest"
    }
  }
}
Same overrides can be used for both dependencies and devDependencies.
If you're using npm version >5 but <8.3.0: edit your package-lock.json: remove the library from "requires" section and add it under "dependencies".
For example, you want deglob package to use glob package version 3.2.11 instead of its current one. You open package-lock.json and see:
"deglob": {
  "version": "2.1.0",
  "resolved": "https://registry.npmjs.org/deglob/-/deglob-2.1.0.tgz",
  "integrity": "sha1-TUSr4W7zLHebSXK9FBqAMlApoUo=",
  "requires": {
    "find-root": "1.1.0",
    "glob": "7.1.2",
    "ignore": "3.3.5",
    "pkg-config": "1.1.1",
    "run-parallel": "1.1.6",
    "uniq": "1.0.1"
  }
},
Remove "glob": "7.1.2", from "requires", add "dependencies" with proper version:
"deglob": {
  "version": "2.1.0",
  "resolved": "https://registry.npmjs.org/deglob/-/deglob-2.1.0.tgz",
  "integrity": "sha1-TUSr4W7zLHebSXK9FBqAMlApoUo=",
  "requires": {
    "find-root": "1.1.0",
    "ignore": "3.3.5",
    "pkg-config": "1.1.1",
    "run-parallel": "1.1.6",
    "uniq": "1.0.1"
  },
  "dependencies": {
    "glob": {
      "version": "3.2.11"
    }
  }
},
Now remove your node_modules folder, run npm ci (or npm install for old version of node/npm) and it will add missing parts to the "dependencies" section.