diff --git a/packages/driver/src/cypress/script_utils.ts b/packages/driver/src/cypress/script_utils.ts
index 7af8b90d8a52..1d6f2ea9dee3 100644
--- a/packages/driver/src/cypress/script_utils.ts
+++ b/packages/driver/src/cypress/script_utils.ts
@@ -17,7 +17,11 @@ const extractSourceMap = ([script, contents]) => {
   const sourceMap = $sourceMapUtils.extractSourceMap(script, contents)
 
   return $sourceMapUtils.initializeSourceMapConsumer(script, sourceMap)
-  .return([script, contents])
+  .catch((_err) => {
+    // if WebAssembly is missing, we can't consume source maps, but it shouldn't block Cy
+    // like in WebKit on Windows: https://github.com/microsoft/playwright/issues/2876
+  })
+  .then(() => [script, contents])
 }
 
 const evalScripts = (specWindow, scripts: any = []) => {
diff --git a/packages/driver/src/cypress/source_map_utils.ts b/packages/driver/src/cypress/source_map_utils.ts
index 24adc14faf74..5b78e1ff3c5c 100644
--- a/packages/driver/src/cypress/source_map_utils.ts
+++ b/packages/driver/src/cypress/source_map_utils.ts
@@ -1,6 +1,5 @@
 import _ from 'lodash'
 import { SourceMapConsumer } from 'source-map'
-import Promise from 'bluebird'
 
 // @ts-ignore
 import mappingsWasm from 'source-map/lib/mappings.wasm'
@@ -12,19 +11,19 @@ const regexDataUrl = /data:[^;\n]+(?:;charset=[^;\n]+)?;base64,([a-zA-Z0-9+/]+={
 
 let sourceMapConsumers = {}
 
-const initializeSourceMapConsumer = (file, sourceMap) => {
-  if (!sourceMap) return Promise.resolve(null)
+const initializeSourceMapConsumer = async (file, sourceMap) => {
+  if (!sourceMap) return null
 
   // @ts-ignore
   SourceMapConsumer.initialize({
     'lib/mappings.wasm': mappingsWasm,
   })
 
-  return Promise.resolve(new SourceMapConsumer(sourceMap)).then((consumer) => {
-    sourceMapConsumers[file.fullyQualifiedUrl] = consumer
+  const consumer = await new SourceMapConsumer(sourceMap)
 
-    return consumer
-  })
+  sourceMapConsumers[file.fullyQualifiedUrl] = consumer
+
+  return consumer
 }
 
 const extractSourceMap = (file, fileContents) => {