The weather was ok at noon at Lords as Middlesex won the toss and decided to field first.
Warwicks scored well early on with Neil Carter hitting fifty from 37 balls before he was out for 68 after hitting ten fours and two sixes.
Ian Bell and Jim Troughton both made thirteen and Tim Ambrose six as Trott played some excellent shots.
With the score on 238, Ian Westwood was out for 33, then Ant Botha was out before Trott, who had hit ten fours from 142, was caught by Berg for 120.
Warwicks ended their fifty overs on 276-7 and Middlesex lost Godleman to the bowling of Carter for 13. Meantime, Phil Hughes had been scoring the runs but he was out for 57 when he was caught by Tim Ambrose off the bowling of Barker.
Ant Both took the wickets of Malan and E.Morgan in two balls but the home side with Owais Shah in good form started to close in on the total.
Jeetan Patel caught Shah when the England man was on 63 for the fifth wicket as Middlesex were placed in a situation where they needed a win to stay in the competition after Scotland who were beaten by Warwickshire on Saturday (video on http://www.cricketscotland.com/) surprisingly beat Kent by 9 wickets.
Warwicks fielding was not as good as it had been in previous matches and the home side were allowed to get extra runs from over throws and poorly timed fielding.
Jon Trott was the exception though and took a fantastic catch to remove Berg (26) and leave the score on 223-6 with 6.5 overs remaining.
Things fluctuated as a no ball by Neil Carter pushed the game towards Middlesex who now needed nine an over. Ben Scott hit a six and then a four off Keith Barker in the 46th over.
Scott had the measure of Carter and left Middlesex needing four to win from the final over. That was it and Dexter (69 not out) hit the winning runs with three balls to spare.
Warwicks play Kent on Wednesday and must win to make the Quarter-Finals. Play is due to start at Edgbaston at 10.45am and you can hear the game on the BBC Coventry & Warwickshire website.
In the test match at Chester Le Street, England beat the West Indies by an innings and 83 runs. Tim Bresnan and Jimmy Anderson mopped up the remaining batsman to give England a 2-0 series win.