Spurs needed a win at White Hart Lane and for arch rivals Arsenal to drop points away to Newcastle to snatch fourth place away from the Gunners.
And for a few moments it seemed to Dawson as though Spurs had done just that when Bale found the top corner of Simon Mignolet's with a stunning 25-yard drive in the 90th minute.
Dawson had assumed that celebrations among sections of Spurs supporters earlier on had meant Arsenal, who had taken a 52nd-minute lead at Newcastle through Laurent Koscielny, were no longer winning at St James' Park.
However, Arsenal had not conceded and their 1-0 win in the north meant Spurs' victory by the same margin was only enough to earn Andre Villas-Boas' men fifth place.
Dawson felt it was going to be a different story following Bale's latest wonder strike though, telling Sky Sports News: "Yeah, I thought that was it (Spurs were in top four) because of the (earlier) celebrations around the ground from our fans.
"I didn't hear anything else, then towards the end after Gaz scored I sort of picked things up that the fans weren't as lively as we would've liked if we'd been in the top four, and you sort of got that feeling towards the end of the game."