use strict;
use warnings;

open FH, "file.txt" or die "can't open file : $!";
my $count=0;
while(<FH>)
{
        my @array=split;
        for (@array)
        {
                if (/a/ && /e/ && /i/ && /o/ && /u/)     # Checking th
+e vowels by using if statement
                {
                # Printing and counting the words
             print "$_\n";
             $count++;
                }
        }
}
        print "No of words: $count";
[download]

I don't think you really mean any other way . . .

#!/usr/bin/env runhaskell
module Main
  where

import System (getArgs)
import qualified Data.ByteString.Lazy.Char8 as L

has_all_vowels :: L.ByteString -> Bool
has_all_vowels word =
  map (\pred -> pred word) predicates == [True,True,True,True,True]
  where
    predicates = (map (\c -> L.elem c) "aeiou")

words_with_all_vowels = filter has_all_vowels . L.lines

main :: IO ()
main = do
  [infile] <- getArgs
  contents <- L.readFile infile
  mapM_ L.putStrLn $ words_with_all_vowels contents

--
-- $ {time ~/vowels.hs /usr/share/dict/words} | wc -l
-- ~/vowels.hs /usr/share/dict/words  0.80s user 0.04s system 99% cpu 
+0.852 total
--    5942
--
[download]

Addendum: Mine doesn't handle AEIOU correctly, but then neither does the OP's.

The cake is a lie.
The cake is a lie.
The cake is a lie.

I asked three things of you:

1. code you've already written
2. explain to us how it should work
3. [explain] where it fails to work for you

You have fulfilled step 1. Maybe now you feel ready to start with step 2?

Your solution isn't bad. There are minor points of improvement (for instance, you dont't really need the variable @array), but why aren't you happy with your code?
-- 
Ronald Fischer <ynnor@mm.st>