Chambers posted the second quickest 60m time in the world this year, 6.58seconds, while Bleasdale triumphed in the pole vault on countback over Anastasia Savchenko.
They were the only wins Britain recorded in Glasgow as they finished with 50 points with Russia and America level on 58 but the former taking the Match courtesy of more individual victories.
Other notable performances included Olympic bronze medallist and British captain Robbie Grabarz finishing second in the high jump and Perri Shakes-Drayton repeating the feat in the 400m.
Dai Green set a new British 600m record of 1:16.22minutes as he beat Andrew Osagie, Duane Solomon the winner, although he was part of the Commonwealth Select team in Glasgow.
Britain beat Germany by a point for third while the Commonwealth Select team were fifth with 40 and British Athletics head coach Peter Eriksson was pleased with what he saw.
"Perri Shakes-Drayton and Robbie Grabarz were fantastic and it is a good start," said Eriksson of his first meet since taking over from Charles van Commenee.
"The athletes are in a very tough training period right now, everyone has been on camps and it is not easy to start up but it was good.
"I am impressed with the youngsters that came in at quite late notice because we had some injuries which gave very short notice to some members of the team."
World indoor triple jump champion Yamile Aldama got Britain up and running with a fifth place while in the second field event Bleasdale produced a leap of 4.60m to win the pole vault.
She failed at 4.70m but took the win as Savchenko herself initially failed at 4.60m while back on the track and Gianni Frankis was fourth in the 60m hurdles.
Serita Solomon went one better in the women's equivalent before Lynsey Sharp was fourth in the 800m, both posting personal bests of 8.26 and 2:05.23 respectively.
Margaret Adeoye and Conrad Williams were second and third in the women's and men's 200m respectively and Grabarz jumped 2.29m for a runners-up spot in the high jump.
Charlie Grice and Laura Muir finished fifth and fourth in the men's and women's 1500m respectively before Shakes-Drayton recorded a personal best in the 400m.
She clocked 52.13seconds which moved her to joint seventh on the all-time British rankings while Dominique Blaize was third in the long jump and Nigel Levine second in the 400m.
Andy Vernon and Anyika Onuora were fourth in the 3000m and 60m respectively before Greene and Osagie's tussle with Solomon in the 600m.
Solomon took the victory while Greene claimed the bragging rights of British record, beating Ikem Billy's 28-year-old mark of 1:17.6, ahead of Osagie.
That left Chambers to claim the final win of the meeting, the multiple European and world indoor medallist clocking 6.58, which is the second fastest time posted so far this year.