Franklin insisted the captain's agreement was made in the spirit of the game.
"Both teams felt we owed it to the occasion to try to make a good game of it," Franklin told BBC Sport.
"We were still under the pump going into Friday's (Saturday NZ time) play but Nick Gubbins and Dawid Malan batted beautifully and forced the hand to start discussions.
"To have the best chance of winning, there would have to be a bit of give and take and I take my hat off to Yorkshire for committing to that because we were committed to it. If Andrew Gale was holding the trophy now, our guys would have walked off with their heads held high. We're very fortunate that we are on the winning side."
The home side began the day on 81-2 and appeared in no particular hurry as they batted through the morning until Gubbins was caught and bowled by Rafiq off a leading edge shortly before lunch.
At that stage they were only 81 runs ahead and after Malan reached his century off 203 balls, a strange interlude followed as Stevie Eskinazi took 27 balls to score his first run, with some spectators baffled by Middlesex's tactical approach.
But with runs and wickets being hard to come by and a draw no good for either side's title hopes, the match took a strange turn after lunch as Adam Lyth and Alex Lees served up a variety of slow deliveries that sparked a run of boundaries.
It was farcical at times, with dot balls receiving ironic cheers from the crowd and boundaries greeted by silence. Although, such action was necessary for Middlesex to set Yorkshire a tempting target.
The innings came to an end when Franklin (30) literally gave his away by soft-batting an unmissable return catch to Lyth, leaving Eskinazi on 78 not out.
